Entropy and Isoperimetry for Linear and
non-Linear Group Actions.
Misha Gromov
June 11, 2008
Contents
1
Concepts and Notation.
1.1 Combinatorial Isoperimetric Profiles I◦ , I◦incr and I◦max . . . . .
1.2 I◦ for Subgroups, for Factor Groups and for Measures on Groups.
1.3 Boundaries |∂|max , ∂µ and the Associated Isoperimetric Profiles.
1.4 G·m -Boundary and Coulhon-Saloff-Coste Inequality . . . . . . . .
1.5 Linear Algebraic Isoperimetric Profiles I∗ of Groups and Algebras.
1.6 The boundaries |∂|max , |∂|µ and |∂|G∗m . . . . . . . . . . . . . . .
1.7 I∗ For Decaying Functions. . . . . . . . . . . . . . . . . . . . . .
1.8 Non-Amenability of Groups and I∗ -Linearity Theorems by Elek
and Bartholdi. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.9 The Growth Functions G◦ and the Følner Functions F◦ of Groups
and F∗ of Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . .
1.10 Multivariable Følner Functions, P◦ -functions and the LoomisWhitney Inequality. . . . . . . . . . . . . . . . . . . . . . . . . .
1.11 Filtered Boundary F∂ and P∗ -Functions. . . . . . . . . . . . . .
1.12 Summary of Results. . . . . . . . . . . . . . . . . . . . . . . . . .
2 Non-Amenability of Groups and Strengthened non-Amenability
of Group Algebras.
2.1
l2 -Non-Amenability of the Complex Group Algebras of nonAmenable groups. . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Geometric Proof that Applies to lp (Γ). . . . . . . . . . . . . . . .
2.3 F- Non-Amenability of the Group Algebras of Groups with Acyclic
Families of Subgroups. . . . . . . . . . . . . . . . . . . . . . . . .
3
3
4
5
6
7
9
10
11
12
14
15
17
19
19
20
21
3 Isoperimetry in the Group Algebras of Ordered Groups.
24
3.1 Triangular Systems and Bases. . . . . . . . . . . . . . . . . . . . 24
3.2 Domination by Ordering and Derivation of the Linear Algebraic
Isoperimetric Inequality from the Combinatorial One. . . . . . . 24
3.3 Examples of Orderable Groups and Corollaries to The Equivalence ◦ ⇔ ∗. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 I∗ -Profiles of Groups with Orderable Subgroups. . . . . . . . . . 26
3.5 I∗ -Profiles of Groups with Orderable Factor Groups. . . . . . . . 27
3.6 Proof of Bartholdi’s I∗ -Linearity Theorem via Orderings. . . . . 27
1
4 Families of Measurable Partitions and their Invariants.
4.1 Four Proofs of the Isoperimetric Inequality for Zk . . . . . . . . .
4.2 Følner Transform F and Schwartz Symmetrization. . . . . . . .
4.3 E◦ -Functions and E◦ -inequalities. . . . . . . . . . . . . . . . . . .
4.4 Slice Removal and Slice Decomposition. . . . . . . . . . . . . . .
4.5 Lower Bounds on F by E◦ . . . . . . . . . . . . . . . . . . . . .
4.6 E◦ -Functions and Group Extensions with and without Distortion.
29
30
31
33
34
35
38
5 Entropy.
5.1 Hölder Inequality via Tensorisation. . . . . . . . . . . . . . . . .
5.2 Entropic Profiles and Stable E◦ Functions of Families of Partitions.
5.3 Shannon Inequalities for the Coordinate Line and Plane Partitions.
5.4 Strict Concavity of the Entropy and Refined Shannon Inequalities.
5.5 On Sliced Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Fiber Products, Fubini Symbols and Shannon Expansion Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Weighted Loomis-Whitney-Shearer and Brascamp-Lieb Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8 Linearized Shearer-Loomis-Whitney Inequalities. . . . . . . . . .
5.9 Orbit and Lieb-Ruskai Inequalities for Representations of Compact Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.10 Reduction Inequality for Orderable Groups Γ. . . . . . . . . . . .
5.11 •-Convexity and the Entropic Følner Functions F• . . . . . . . .
5.12 Entropic Poincaré Inequalities. . . . . . . . . . . . . . . . . . . .
39
40
41
43
44
47
6 Lower Bounds on F∗ for Groups with Large Orderable Subgroups.
6.1 Expanding Subgroups Inequality. . . . . . . . . . . . . . . . . . .
6.2 Distorted and Undistorted Actions on l2 and on Decaying p-adic
Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Isoperimetry with ”Large” H ⊂ Γ. . . . . . . . . . . . . . . . . .
49
50
53
54
56
57
58
59
59
60
62
7 Linear Algebraic Entropic Inequalities for Normal Group Extensions.
64
7.1 Vertical Families L of Linear Subspaces and L-Entropy. . . . . . 64
7.2 L-Entropic Følner Functions. . . . . . . . . . . . . . . . . . . . . 66
8
Examples of Group Algebras with Fast and with Slow Growing
Følner functions.
68
8.1 An Upper Bound on I∗ for Split Extensions. . . . . . . . . . . . 68
8.2 Amenable Groups with Fast Growing Følner Functions. . . . . . 70
9
E◦ -Functions and Isoperimetric Inequalities for Wreath Powers
of Partitions and Groups .
9.1 Wreath Powers of Sets. . . . . . . . . . . . . . . . . . . . . . . .
9.2 Følner Functions for Iterated Wreath Product of Groups. . . . .
9.3 Infinite o- and ↑-Products. . . . . . . . . . . . . . . . . . . . . . .
2
72
72
74
76
10 Brunn-Minkowski inequalities.
78
10.1 Asymptotic Brunn-Minkowski inequalities for Discrete Groups. . 79
10.2 Linearized BM-inequalities for Biorderable Groups. . . . . . . . . 81
10.3 BM-Inequality for Fast Decaying Functions on Γ. . . . . . . . . . 82
11 Bibliography.
84
Abstract
We prove inequalities of isoperimetric type for groups acting on linear
spaces and discuss related geometric and combinatorial problems, where
we use the Boltzman entropy to keep track of the cardinalities (and/or
measures) of sets and of the dimensions of linear spaces.
1
1.1
Concepts and Notation.
Combinatorial Isoperimetric Profiles I◦ , I◦incr and I◦max .
Consider a measure space X (e.g. a discrete set with all points carrying unit
weights) where the measure of a subset Y ⊂ X is denoted by |Y | (that is the
cardinality for the discrete spaces with unitary atoms).
If g : X → X is an invertible measure preserving map we define the gboundary of Y ∈ X as the set difference
∂g (Y ) = g(Y ) \ Y.
If G is a set of such maps g : X → X, define the (joint exterior) G-boundary
joint
∂G (Y ) = ∂G
(Y ) ⊂ X of Y ⊂ X as the union
∂G (Y ) = ∪g∈G ∂g (Y ) = (G · Y ) \ Y
and observe that
|∂G (Y )| = |(G · Y )| − |Y |,
provided G 3 id.
An isoperimetric inequality in this context is the inequality of the form
(◦)
|∂G (Y )| ≥ I(|Y |),
where (◦) is supposed to hold for all subsets Y ⊂ X of finite measure (just finite
subsets in the discrete case) with a given function I = I(r) of real argument
r ≥ 0.
The combinatorial isoperimetric profile of X with respect to G is the maximal
function, denoted
I◦ (r) = I◦ (r; X, G)
for which all Y satisfy (◦).
Combinatorial Isoperimetric Problem: Evaluate I◦ and find extremal subsets
Y ⊂ X that minimize |∂(Y )| among all Y of given cardinality/measure. (The
classical solution to this problem in the Euclidean plane is given by the round
discs Y bounded by circles ∂(Y ).)
Γ-Spaces X. This means that X is a Borel measure space with a group Γ
acting from the left on X by measure preserving transformations. For instance,
3
X may be equal to Γ with the left action, where Γ is a countable (or locally
compact) group with a given finite (compact) subset G ⊂ Γ that is usually (but
not always) assumed to generate Γ.
I◦ (r) for real numbers r. It is convenient to have functions of positive integer
arguments extended to all r ∈ R+ ; we tacitly assume such extension with the
functions I◦ (r) (and similar functions we shall meet later on ) being constant
or linear on the integer intervals (i, i + 1).
Asymptotic Equivalence. If Γ is an infinite finitely generated group then
the combinatorial isoperimetric profiles for different finite generating subsets
G, G0 ⊂ Γ are asymptotically equivalent, i.e. their ratio is bounded away from 0
and ∞.
In fact if G ⊂ G0 ⊂ G·m , where G·m ⊂ Γ denotes the set of the words
in the letters g ∈ G of length ≤ m then the corresponding profiles satisfy
I ≤ I 0 ≤ m · |G0 | · I.
This equivalence class is then regarded as the combinatorial isoperimetric
profile of Γ that does not depend on G.
Sublinearity of I◦ . The function I◦ (r) = I◦ (r; Γ, G) is bounded by |G|r and
it satisfies the inequality
(+)◦
I◦ (r1 + r2 ) ≤ I◦ (r1 ) + I◦ (r2 )
for all infinite groups Γ and finite subsets G ⊂ Γ.
Non-monotonicity of I◦ and I◦incr . The profile I◦ (r) of a group Γ is not
always monotone increasing in r. For example, I◦ (r ≥ |Γ|) = 0 for finite groups
Γ. If we insist on the monotonicity, we define
I◦incr (r) = infs≥r I◦ (s),
that is the maximal monotone increasing minorant of I◦ .
Possibly, the ratio I◦ (r)/I◦incr (r) is bounded for all infinite finitely generated
groups.
Remark. The notion of the isoperimetric profile for groups was introduced
by Anatoly Vershik [29] in the (essentially equivalent) language of the Følner
functions defined below.
1.2
I◦ for Subgroups, for Factor Groups and for Measures
on Groups.
The inequality (+)◦ implies that the G-isoperimetric profile of a free action of
an infinite group (Γ, G) on an arbitrary space (set) X equals the G-profile of Γ,
since the boundary of a subset in X equals the disjoint union of the boundaries
of the orbits. Consequently,
the profile of every group extension (Γ1 ⊃ Γ, G1 ⊃ G) is bounded from below
by the profile of (Γ, G),
I◦ (r; Γ1 , G1 ) ≥ I◦ (r; Γ, G).
Also
I◦ decreases under epimorphisms of infinite groups .
4
Namely, let h : Γ → Γ be a surjective group homomorphism, G a finite
subset in Γ and let G = h(G) ⊂ Γ be the image of G in Γ. Then
I◦ (r; Γ, G) ≥ I◦ (r; Γ, G).
Proof. Given Y ⊂ Γ, let Y m ⊂ Γ, m = 1, 2, ..., be the subset of points with
at least m preimages in Y . Clearly,
(+)
X
|∂G (Y )| ≥
|∂G (Y m )|
m=1,2,...
and the subadditivity applies, since
X
|Y m | = |Y |.
m=1,2,...
Subgraphs of Functions and their Boundaries. Given a positive function f (x)
on a space X, define the subgraph Y f ⊂ X × R+ as the set of pairs (x, r), such
that f (x) ≤ r. Then, for a set G of transformations of X, let
|∂G (f )| =def |∂G (Y f )|
for the obvious action of G on X × R+ .
If the function f is integer valued, then |∂G (f )| equals the sum of the Gboundaries of its levels Ym =def {f (x) = m} ⊂ X, m = 0, 1, 2, ...,
X
|∂G (Ym )|;
|∂G (f )| =
m
thus one obtains the isoperimetric lower bound on |∂G (f )| by the l1 -norm
||f ||l1 =def |Y f | of f ,
|∂G (f )| ≥ I◦ (||f ||l1 ; G),
provided I◦ is subadditive. (A similar inequality for non-integer valued functions
needs extra entropic data on f , see 5.12).
This applies, in particular to the function f (γ) = |h−1 (γ)| for the above h :
Γ → Γ, where the essential feature is the (obvious) inequality |∂G (f )| ≤ |∂G (Y )|.
Generalization to maps between Γ-Spaces. Let X and X be spaces acted
upon by Γ and X → X be a Γ-equivariant map, where a basic example is
X = Γ and X = Γ/Γ0 . The function I◦ (r; X) is not always subadditive, e.g. for
some quotients Γ/Γ0 where Γ is the free group F2 . If this is the case we work
with the maximal subadditive minorant I◦sb.ad (r) of I◦ (r) and observe with the
above argument that
I◦sb.ad (r; X) ≥ I◦sb.ad (r; X).
1.3
Boundaries |∂|max , ∂µ and the Associated Isoperimetric
Profiles.
The boundary of a Y ⊂ X can be regarded as a function of g ∈ G for g 7→ ∂g (Y )
and, besides taking the union of these g-boundaries, one may take the supremum
5
of their cardinalities and, thus, define |∂|max (Y ) with I◦max (r, Γ, G) denoting the
corresponding profile.
P
Another useful boundary is the µ-boundary |∂µ (Y )| = g µ(g)|∂g (Y )| for a
measure µ on G, with I◦ (r; , Γ, µ) for the associated profile.
If µ equals the Haar measure of Γ restricted to a subset H ⊂ Γ ( this H may
be different from the original G when it comes to applications ), this boundary
is denoted |∂ΣH |(Y ). If the group is discrete with the Haar measure having
unitary atoms, then
|∂ΣH |(Y ) = |Y | · |H| −
(ΣH)
X
|Y ∩ g(Y )| = |Y | · |H| −
X
|y1 y2−1 |H ,
Y ×Y
g∈H
where |y1 y2−1 |H denotes the number of g ∈ H, such that g(y1 ) = y2 .
All these boundaries lead to the equivalent isoperimetric profiles for individual G but they behave differently for families Gi , where |Gi | → ∞.
The essential properties of these boundaries are quite similar. For example,
all of them except for |∂|max are slice-wise additive for actions of Γ on X with
several orbits (slices) S:
P
|∂| of every Y ⊂ X equals the sum S |∂|(Y ∩ S).
In what follows, everything applies to all types of boundaries unless stated
otherwise.
The boundary |∂|max , albeit non-additive, is more convenient than ∂ joint for
Cartesian products of groups (and spaces) as is seen in the following
Z2 -Example. Consider the group Γ = Z2 = Z × Z with the standard generators g1 and g2 . Then every Y ⊂ Z2 , obviously, satisfies |Y | ≤ |∂|max (Y )2 ,
where the extremal sets are the n × n-squares. But if we turn to the joint exterior boundary,
we see that the diamond, the square rotated by 90◦ , has this
√
boundary 2-times smaller than the square, while the cardinalities of the sets
in Z2 do not change under the rotations away from the boundary.
1.4
G·m -Boundary and Coulhon-Saloff-Coste Inequality .
Start with the following triangle inequality.
(∂g1 ·g2 )
|∂g1 ·g2 (Y )| ≤ |∂g1 (Y )| + |∂g2 (Y )|,
for every pair of invertible transformations g1 and g2 of X.
To see this clearer, look at the ”X-cube” Z = 2X , i.e. the set of all finite
subsets Y ⊂ X with |Y | < ∞ where the distance is given by the the maximum
of the measures (or the cardinalities) of the differences of subsets (representing
non-symmetric metric on the set of subsets) and denoted
|Y1 − Y2 | =def max(|Y1 \ Y2 |, |Y2 \ Y1 |).
Since transformations g of X isometrically act on Z and ∂g (Y ) = |Y − g(Y )|,
the inequality (∂g1 ·g2 ) follows from the triangle inequality in 2X .
It follows that
The boundary of every subset Y ⊂ X with respect to the m-ball G·m ⊂ Γ
satisfies
6
1 max
|∂| ·m (Y )| ≤ |∂|max
G (Y ).
m G
(∂G·m )
On the other hand, if the action of Γ on X is free, the above identity ΣG ,
for H = G·m implies
|∂|max
G·m |(Y ) ≥ (1 −
|Y |
)|Y |
|G·m |
and yields the following inequality due to Coulhon-Saloff-Coste [4] .
CSC Displacement Inequality
|∂|max
G (Y ) ≥
(DIS)
|Y |
1
(1 − ·m )|Y |
m
|G |
for all Y ⊂ Γ and all m = 1, 2, ....
Inverse ∂G·m -Inequality. In most (if not all) known cases there is a converse
inequality saying that the function I(r, m) = I◦max (r; G·m ) is, roughly, linear in
m, i.e. I(r, m) ∼ m · I(r, 1) for m ≤ r/2I(r, 1).
It is unclear if this is true in general and what are more general constrains
on possible (Γ-invariant) metrics on the orbits Γ(Y ) ⊂ 2Γ for various Y ⊂ Γ.
On the other hand, (DIS) admits the following (strengthened) generalization.
Intersection Inequality. A subset H ⊂ Γ, is called (right) G-connected if every two points h0 and h in H can be joint by a chain of points (h0 , h1 , ..., hi , ..., hk =
h), such that hi = hi−1 gi for some gi ∈ G and all i = 1, 2, ..., k. Observe that the
subsets H = G·m ⊂ Γ are G-connected for all m = 1, 2, ...; moreover, if G−1 = G
and G 3 id, then every two points in G·m are contained in a G-connected subset
of cardinality ≤ 2m.
If a finite subset H ⊂ Γ is G-connected for a given G ⊂ Γ, then all measurable subsets Y of an arbitrary Γ-space X (with the action written ”on the left”:
x 7→ γx) satisfy
\
|
h(Y )| ≥ |Y | − (|H| − 1)|∂|max
G (Y ).
h∈H
Proof. Observe that |∩h∈H h(Y )| = |Y | for |H| = 1 and proceed by induction
on |H| for |H| > 1. To do this, decompose H = H 0 ∪ {h0 g} for a G-connected
subset H 0 ⊂ H with |H 0 | = |H| − 1 and some h0 ∈ H 0 and g ∈ G and notice
that the intersection inequality is stable under the left translations of H.
0
Thus, we pass from H to h−1
0 H and arrive at the case where h0 = id ∈ H
0
and H = H ∪ {g}, and where, obviously,
\
\
\
\
|(
h(Y )) g(Y )| ≥ |
h(Y )| − |(Y \ g(Y ))| = |
h(Y )| − |∂g (Y )|.
h∈H 0
1.5
h∈H 0
h∈H 0
Linear Algebraic Isoperimetric Profiles I∗ of Groups
and Algebras.
Let L be a linear space over some field F and G a set of linear operators L → L
, where in our our examples these operators are invertible, injective as well
7
as surjective. We define the (joint exterior) G-boundary of a linear subspace
D ⊂ L as the quotient space
joint
∂G (D) = ∂G
(D) = (G ∗ D)/D,
where G ∗ D ⊂ L denotes the linear span of the subspaces g(D) for all g ∈ G
and where A/B =def A/A ∩ B. Set
|∂G (D)| = rankF ∂G (D)
(that equals rankF (G ∗ D) − rankF (D) if G 3 id) and define the isoperimetric
profile of L with respect to G as the maximal function I∗ such that all finite
dimensional subspaces D ⊂ L satisfy the following ”isoperimetric inequality”,
(∗)
|∂G (D)| ≥ I∗ (|D|),
where |D| stands for rank(D) = rankF (D).
Main Examples: Group Algebras and their Extensions. Let L = A be the
group algebra of Γ over some field F regarded as the vector space of the F-valued
functions on Γ with finite supports. The group Γ linearly acts on A and, for a
given subset G ⊂ Γ, one may speak of the G-boundary ∂G in A as well as in Γ.
Supports of Linear Spaces and the Implication (∗) ⇒ (◦). The support of
a linear space D of functions on Γ is, by definition, the complement to the set
where all functions from D vanish. If |supp(D)| = |D| then D equals the space
FY of all functions on Y = supp(D) and, obviously, |∂G (D)| = |∂G (Y )|. Thus,
I◦ ≥ I∗ .
The reverse inequality does not always hold:
there are finitely generated groups (with torsion), where the two profiles are
not equivalent; for example I∗ (r) can be bounded while r/I◦ (r) grows arbitrarily
(depending on Γ) slowly (see 8.1).
It is possible, albeit unlikely, that I∗ (r) ∼ I◦ (r) for groups without torsion.
(This is similar to the Kaplansky conjecture saying that the group algebras of
torsionless groups have no divisors of zero).
Sublinearity and Monotonicity of I∗ of Group Algebras. Clearly,
(+)∗
I∗ (r1 + r2 ) ≤ I∗ (r1 ) + I∗ (r2 )
for infinite groups Γ; therefore,
the profile of a group algebra of an infinite group may only increase under a
group extension.
Also we shall prove in 3.5 that
I∗ decreases under surjective group homomorphisms when the target group
is left orderable.
It remains unclear to what extend this remains valid in general but we shall
see in 7.2 that some entropic isoperimetric profile is always monotone decreasing
under surjective group homomorphisms of infinite groups.
Remark. The profile I∗ is residually monotone: if Γi ⊂ Γ is a decreasing sequence of normal subgroups, such that ∩i Γi = {id}, then I∗ (r; Γ) ≥
8
limi→∞ I∗ (r; Γ/Γi ), since the push forward homomorphism of the group algebra
A of Γ to the algebra Ai of Γ/Γi is injective on each finite dimensional subspace
D ⊂ A for large i = i(D). (This remains true for the algebra of l1 -functions on
Γ).
About dependence of I∗ on F. The action of G 3 id acting on r functions
X → F is described by the following infinite system of linear equations in
unknowns fjxγ = fj (γ(x)), indexed by (x ∈ X, γ ∈ G, j = 1, 2, ..., r),
LIN
fjγ(x)id = fjxγ
The bound I∗ (r) ≤ a means that there is a linear space D+ of solutions of
LIN in functions with finite supports, such that |D+ | ≤ r + a while the map
from D+ to the space of r-tuples of functions on x given by fjxγ 7→ fj (x) has
rank r.
Thus, by the Lefschetz principle,
if F is algebraically closed, then the profile I∗ for F-functions with finite
supports on any Γ-space X depends only on the characteristics of F
( I∗ may depend on F, e.g. for finite groups, but the extent of this dependence
is unclear.)
1.6
The boundaries |∂|max , |∂|µ and |∂|G∗m .
One defines |∂|max and |∂|µ as in the combinatorial case with the corresponding notation I∗max (r; Γ, G) and I∗ (r; Γ, µ) for the isoperimetric profiles and one
observes that
1 max
|∂| ·m (D)| ≤ |∂|max
G (D)
m G
(∂G∗m ).
Thus,
1 max
I
(r; G·m ),
m ∗
where the triangle inequality applies to the metric on the set (Grassmannian)
of subspaces D in L defined by
I∗max (r; G) ≥
|D1 − D2 | =def
1
(|D1 /D2 | + |D2 /D1 |).
2
Furthermore,
|
\
h(D)| ≥ |Y | − |H| · |∂|max
G (D).
h∈H
for H as in the Intersection Inequality from 1.4 and finite dimensional subspaces
D in linear Γ-spaces.
Remarks. (a) There is a larger set of operators, denoted G?m ⊃ G·m , which
is generated by m operations of additions as well as compositions of linear maps
g ∈ G. Namely, start from
9
G?1 =
[
cG
c∈F
and then set
G?m =
[
((G?m1 · G?m2 ) ∪ (G?m1 + G?m2 )) .
m1 +m2 ≤m
Here again,
I∗max (r; G) ≥
1 max
I
(r; G?m ).
m ∗
and
1
|∂|µ (D),
m
for a probability measure µ0 on G depending on a given probability measure µ
on G?m .
(b) Boundary defined with Non-invertible Operators. The G-boundary is
supposed to measure the degree of (non)invariance of a subspace D under the
operators g ∈ G and if (some of) the g ∈ G are non-injective on D one may
need to include into the G-boundary the kernels of these g.
|∂|µ0 (D) ≥
1.7
I∗ For Decaying Functions.
Let the underlying field F come with a norm, e.g. F equals the field of real,
R, complex C, or p-adic Qp numbers. Then every decaying F-function f on a
countable set X defines an indexing of X by positive integers i ∈ N+ , for which
the sequence of norms, s(i) = ||f (xi )|| is (non-strictly) monotone decreasing.
If all functions f in a finite linear space D of functions on a set X (e.g.
on X = Γ decay super-exponentially i.e. the corresponding sequences satisfy
C i s(i) → 0 for all C > 0 and i → ∞, then one can replace (approximate) D by
a linear space D0 of functions with finite supports on X such that |D0 | = |D|
and |∂G (D0 )| = |∂G (D)| for a given finite set G of transformations of X.
Sketch of the Proof. The determinants of the (k × k)-minors of the matrix
corresponding to the system LIN in 1.5 are integers (exponentially) bounded by
|G|k . The solutions of the system LIN in the variables fx , now indexed by i ↔ xi ,
can be obtained by extending the solutions of the sub-systems for fjγ (xi ), j ≤
jN → ∞ and such extension amounts to solving some non-homogeneous system
associated with LIN, where the non-homogeneity takes care of the |D| = r
condition by prescribing the values of the solutions (functions) at certain points
in X. The rate of decay (as well as of growth) of such extensions can not be
(much) faster than |G|jN and then the existence of a faster decaying solution
implies that for a finitely supported one.
(The “rate of decay” on linear subspaces D is seen with the help of suitable
bases in D, where the superexponentiality is apparently needed. We leave the
details to the reader, since we do not use such decaying solutions in this paper).
Questions. Does the isoperimetric profile of the space of decaying (i.e. converging to zero at infinity) functions with values in an ultrametric field always
equal (or, at least, is equivalent) to I∗ for functions with finite supports? Does
a similar equality (or at least, equivalence) hold for the Hilbert spaces of square
summable real and complex valued functions?
10
We shall prove the equality for the free Abelian groups and the equivalence
for the non-virtually nilpotent polycyclic groups in 6.2.
Non-decaying functions, Representations, Finite Groups, etc. We are mostly
concerned in this paper with the regular representation of an infinite group Γ
where we try to ”linearize” combinatorics of finite subsets in Γ.
Other infinite dimensional, say irreducible, representations L of Γ, e.g. realized by spaces of (non-decaying) functions on Γ, may have ”good Isoperimetry”
for ”a non-combinatorial reason”, e.g. if they do not contain, in some weak
sense, ”bad” sub-representations.
Also |∂(D)|/|D| may be bounded from below in terms of the distance from
|D| to ”bad” invariant subspaces. e.g. those of dimensions close to |D|,where
many examples come from (families of) finite groups, such as GL(k; Fp ) where
suitable Følner functions, probably, behave ”nicely” for k, p → ∞.
1.8
Non-Amenability of Groups and I∗ -Linearity Theorems by Elek and Bartholdi.
A countable group is called non-amenable if it satisfies the linear isoperimetric
inequality with some finite subset G ⊂ Γ (where one may assume that G generates Γ if the group is finitely generated), i.e. if the function I◦ (r) is the maximal
possible up to the asymptotic equivalence: I◦ (r) ∼ r, which amounts here to
I◦ (r) ≥ cr for c > 0; otherwise the group is called amenable.
The basic examples of non-amenable groups are the free groups on two or
more generators and the groups containing such free groups as subgroups. Also,
there are many non-amenable groups without free subgroups, e.g. non-amenable
torsion groups.
On the other hand all solvable (e.g. Abelian groups) are amenable.
Remark. All known infinite amenable groups, in particular the groups of
subexponential growth, either contain arbitrarily large finite subgroups or they
have ”many” commuting elements but no general result of this kind is known.
Here is a test question.
Let Z(γ0 ) ⊂ Γ for a given γ0 ∈ Γ consist of those γ ∈ Γ, where some powers
γ0m0 and γ m1 commute. Does every infinite amenable group Γ contain a nonidentity element γ0 ∈ Γ, such that |Z(γ0 ) ∩ G·n |/n → ∞ for n → ∞, where
G ⊂ Γ is a finite generating subset?
An action of an algebra A on a linear space L is called non-amenable or
I∗ -linear, if its isoperimetric profile I∗ (r) defined with some finite subset of
operators G ⊂ A is equivalent to r. If L equals the algebra itself then the
algebra is called I∗ -linear or non-amenable. (Unlike the group theoretic case, it
is unclear what is the true property that divides algebras into ”amenable” and
”non-amenable” categories; the less obliging terminology seems appropriate.)
It is obvious that if Γ is amenable then so is each of its linear actions and
the converse is known in the following cases.
I∗ (F2 )- Linearity Theorem. The group algebra of the free group F2 , and
hence, of every group containing F2 is I∗ -linear for all fields F. (Gabor Elek
[7]; see 2.3. for a strengthening of this result).
l2 - Non-amenability Theorem. The action of every non-amenable group
Γ on the complex Hilbert space l2 (Γ) is I∗ -linear. (Gabor Elek [6]; see 2.2 for
the case of lp , p 6= 2).
11
I∗ -linearity Theorem. The group algebras of all non-amenable groups
over all fields F are I∗ -linear. (Laurent Bartholdi see [2] and see 3.6 for a
combinatorial rendition of Bartholdi’s proof; if char(F) = 0, this follows from
the l2 -theorem.)
Question. Is the action of a non-amenable group Γ (e.g. for Γ = F2 ) on
the space of decaying F-valued functions on Γ, where F is a normed field (e. g.
the field of real numbers or the field of formal power series over a finite field),
I∗ -linear?
1.9
The Growth Functions G◦ and the Følner Functions
F◦ of Groups and F∗ of Algebras.
The growth function G◦ (n), n = 1, 2, ..., of a group Γ relative to a given generating set G ⊂ Γ is the number of elements in Γ representable by the words of
length≤ n in the letters g ∈ G. That is G◦ (n) = |G·n | for
G·n =def G
· ... · G} ⊂ Γ.
| · G{z
n
The growth function is at most exponential for finite sets G, since |G·n | ≤
n
|G| .
The Følner function F◦ = F◦ (n), n = 1, 2, ..., of (Γ, G) is equal, by definition,
to the minimal cardinality of a (Følner) subset Yn ⊂ Γ, such that |∂G (Yn )| ≤
1
n |Y |.
This can be expressed entirely in terms of I◦ : define, for an arbitrary function
I, the Følner function associated to I by the rule:
the value of the function FI at n equals the minimal N , such that I(N ) ≤
N/n and observe that FI (n) = F◦ (n; Γ) for I(r) = I◦ (r; Γ). In other words, FI
is the inverse function to the minimal monotone increasing majorant M (r) of
the function r/I(r). Thus, returning from F back to I, gives us r/M (r) ≤ I(r)
instead of I(r) where the equality takes place if and only if if r/I(r) is monotone
increasing. (We disregard the minor ambiguity which arises for non-strictly
monotone functions.)
Unlike the isoperimetric profiles I(r) that grow at most linearly and often
are concave functions, the Følner functions of infinite groups grow faster than
linearly and are rather convex than concave, where the rate of their growth
translates into how slowly the corresponding isoperimetric profiles deviate from
non-constant linear functions.
One may regard the map Y 7→ ∂Y as an ”operator” on the ”space of
sets” with the ”|...|-norm” and think of the corresponding function M (r) =
sup|Y |≤r |Y |/|∂Y | as the ”norm” of the ”inverse operator” ∂ −1 where this ”norm”
is not a number but rather a function depending on r = |Y |, since ∂ is a ”nonlinear operator”.
It is also helpful to think of the ratio n = |Y |/|∂(Y )| as a linear size (diameter) of Y and the Følner function as the volume (of the minimal Y with given
diameter,) depending on the diameter.
It is clear that FcI (n) = FI (cn) for all c > 0. Thus, for example, the G·m 1
I◦ (r; Γ, G·m (see 1.4) translates to
inequality I◦ (r; Γ, G) ≥ m
(n/m),
F◦ (n; Γ, G) ≥ F◦ (n/m; Γ, G·m ).
12
and the (DIS)-inequality from 1.4 applied to the minimal m for which |G·m | ≥
2|Y |, yields
Couhlon-Saloff-Coste Isoperimetric Inequality.
(F◦ G◦ )
F◦ (n; Γ, G) ≥
1
n
G◦ ( ; Γ, G).
2
2
This inequality (that improves upon an earlier result by Varopoulos [28]
with a rather elaborate proof exploiting random walks in groups) implies, for
instance, that
the groups of exponential growth have at least exponential Følner functions.
But unlike G◦ (n), the Følner functions F◦ (n), of some (amenable!) groups
may grow (much) faster than exponentially as was conjectured by A. Vershik in
the late 60s and confirmed in [27] and in [9]. (We meet such groups in the later
sections.)
The useful notion of equivalence between two Følner functions is different
from that for I:
we write F ≺ F 0 if F (n) ≤ F 0 (Cn) for some constant C > 0 and F ∼ F 0 if
F ≺ F 0 as well as F 0 ≺ F and we apply the same notion of equivalence to the
growth functions.
Examples (a) The Coulon-Saloff-Coste inequality gives the correct asymptotic for F◦max (n) ∼ nk for nilpotent (e.g. Abelian) groups (as was earlier proved
by Pansu for the Heisenberg group and by Varopoulos in general).
Remark. There is a significant discrepancy between the lower bound F◦max (n) ≥
1
1
For instance, if Γ = Zk , then
2 G◦ ( 2 n) and the available upper bounds.
1
k
k
G◦ (n) = k! (2n) + o(n ), while the (obvious) upper bound F◦max (n) ≤ nk
is provided by the n-cubes Cn ⊂ Z k , where |Cn | = nk and |∂ max |(Cn ) = nk−1 .
(In fact, the cubes are the extremal sets in Z k and F◦ (n; Zk ) = nk , see 4.1).
Question. What is the asymptotic of F◦ (n; Γ(k1 , k2 )) for the free nilpotent
groups Γ(k1 , k2 ) on k1 generators and of the nilpotency degree k2 for n, k1 , k2 →
∞?
(b) A group is non-amenable if and only if F◦ (n) = ∞ for large n.
(c) F◦ (n; Γ) is bounded if and only if Γ is finite, where it equals |Γ| for n > |Γ|,
while infinite groups have F◦ (n) n where the equivalence F◦ (n; Γ) ∼ n implies
(by an easy argument) that the group is commensurable with Z.
Følner functions F max , F∗ , etc. Since the definition of F◦ makes sense for an
arbitrary function I(r) instead of I◦ one can define F∗ associated to the profiles
of group algebras as well as the Følner functions for |∂|max , in groups and linear
spaces alike.
We shall introduce in 4.2 certain operations on functions, called Følner transforms that correspond to some group theoretic constructions, e.g. Cartesian
product and wreath products (see 9.1) and show that these are particularly
nicely behave on the class of what we call •-convex functions where the notion of the •-convexity incorporates the Shannon inequality for the entropies of
random variables.
The first instance where one sees the difference between F∗ and F◦ appears
for finite groups:
the Følner functions of the group algebras of finite groups Γ are F∗ (n; Γ) = 1
for all n = 1, 2, ..., since the 1-dimensional space of constant functions has
13
zero boundary. This leads (see 8.1) to examples of ”very large” groups where
F∗ (n; Γ) ∼ n.
On Growth of Algebras. Let L be a linear space with a set G of bijective
operators acting on it, e.g. the group algebra of Γ. Then the growth function
of L, G is defined with some vector l ∈ A as
G∗ (n, L, l, G) =def |G·n ∗ l|
where G·n ∗ l denotes the linear span of the orbit G·n (l) ⊂ L.
If the group algebra A of Γ generated by G acts freely or without torsion
on L, i.e. a(l) = 0 only if a = 0 and l = 0, then this growth function equals
that of Γ. In general, it may be smaller than G◦ (n; Γ, G) and one may define
Ginf
∗ (n, L, G) as the infimum of G∗ (n; L, l, G) over all l 6= 0 in L.
It is unknown (Kaplansky conjecture) if the freedom condition is satisfied
by the group algebras of the groups without torsion and it is unclear how much
G∗ (n; L, l, G) may depend on l in general.
Question. When is G∗ (n, L, l, G) asymptotically equivalent to Ginf
∗ (n, L, G)?
Observe that the (obvious) linear algebraic counterpart of the (DIS)-inequality
(see 1.4) reads as follows,
if an algebraic family H of linear operators a acting on a linear space X
over an algebraically closed field has no zero vector in a given linear subspace
Y ⊂ X, i.e. a(y) 6= 0 for a 6= 0 and y 6= 0, then
|∂|max
H (Y ) ≥ dim(Y )(1 − dim(Y )/dim(H)).
However, this does not match, at least not in an obvious way, the (G∗m )inequality for the linear actions.
It remains unclear if there is a bound on the growth of the algebra by the
Følner function, say for the torsion free (i.e. without divisors of zero) algebras.
1.10
Multivariable Følner Functions, P◦ -functions and the
Loomis-Whitney Inequality.
The multivariable Følner function F◦ (ni ; X, Gi ) for several sets of transformations Gi acting on X is defined as the maximal N such that every Y ⊂ X
with
|∂Gi (Y )| ≤
1
|Y |
ni
has
|Y | ≥ N
and similarly, one defines multivariable F∗ for linear actions.
These functions reduce to the ordinary F◦maxi (n) and F∗maxi (n) at n1 =
n2 = ... = n, where maxi refers to |∂| = maxi |∂Gi |.
The P◦ -function for Orbit Partitions. Let X be acted upon by groups Γi ,
i = 1, 2, .., k, and denote by Γ◦ the intersection ∩i Γi . Consider subsets Y ⊂ X
that have finite (or finite measure) images Y /Γ◦ in the quotient space X/Γ◦ and
define P◦ (ni ; X/Γi ) as the minimal N such that the inequalities
14
|Y /Γi | ≤
1
|Y /Γ◦ |, i = 1, 2, ..., k, imply |Y /Γ◦ | ≥ N.
ni
Observe that this P◦ -function, unlike I◦ and/or F◦ , depends only on the Γi orbit partitions structure in X, rather than on the full Γ-structure in X.
If Γ◦ = {id}, while the the Γi -orbits are all infinite, then P◦ (ni ) ≤ F◦ (ni ),
since each infinite Γi orbits that meets a finite subset Y ⊂ X contributes a point
to the Γi -boundary of Y .
Basic Example: Loomis-Whitney Theorem. (see 5.7). The P◦ -function for
the family of the k partitions of Rk by the lines parallel to the k coordinate axes
is,
Y
P◦ (ni ) =
ni .
i
In other words,
among all subsets Y ⊂ Rk with given measures of the projections to the
k coordinate hyperplanes, the maximal measure is achieved by the rectangular
solids (and all subsets obtained from them by measurable transformations of Rk
preserving the coordinate line partitions).
It follows that
the multivariable Følner function of the free Abelian group Zk with respect
to its k generators gi ∈ Zk (i.e. Gi = {gi }) is
F◦ (n1 , n2 , ..., nk ) = n1 · n2 · .... · nk .
Remark. We shall explain in 5.3 (this was pointed out to me by Noga Alon)
how [LW ] follows from the Shannon inequality for the entropies of random
variables, where the latter, in the case of Rk , is (essentially) equivalent to the
logarithmic Sobolev inequality.
1.11
Filtered Boundary F∂ and P∗ -Functions.
F∗ (ni ) and P∗ (ni ) for linear actions. The definition of the multivariable Følner
function F∗ for groups Γi acting on a linear space L is identical to that of F◦ ,
while P∗ needs a replacement of |Y /Γ| by a suitable Γ-rank (dimension), denoted
|D : Γ| for linear subspaces D ⊂ L.
There are several candidates for P∗ corresponding to different notions of the
rank of a linear Γ-space. We start with a notion of a boundary associated to
the Krull dimension.
F∂-boundary. Given a group Γ (or, more generally, an algebra A) acting
on a linear space L, consider a Γ-invariant ascending filtration F = {0 = L0 ⊂
L1 ⊂ L2 ⊂ ... ⊂ L}. Then, for a given linear subspace D ⊂ L consider the
subspaces Di = D ∩ Li and let ∂F (D) denote the number of those i = 1, 2, ...
for which the subspace Di /Di−1 ⊂ Li /Li−1 is not Γ-invariant. Clearly, this
boundary minorizes the Σ-boundary ∂ΣH (D) for an arbitrary generating subset
H in Γ.
Then we define
F∂Γ (D) = sup ∂F (D)
F
for all Γ-invariant filtrations F in L and, given groups Γi acting on L, one defines
the ”filtered” isoperimetric profile associated to the boundaries |F∂Γi (D)| and
15
the corresponding multivariable Følner function. (The corresponding boundary
in the combinatorial framework of Γ acting on X counts the number of the
Γ-orbits meeting a Y ⊂ X that are not contained in Y .)
Let the group Γ be amenable and H be an exhaustion by an increasing
sequence of (Følner) subsets Hj ⊂ Γ with |∂Hj |/|Hj | → ∞. If two finite
dimensional subspaces D1 , D2 ⊂ L, have equal Γ spans Γ ∗ D1,2 ⊂ L, then their
Hj -spans Hj ∗ D1,2 have the same asymptotic growth,
|Hj ∗ D1 |/|Hj ∗ D2 | → 1
for j → ∞.
On the other hand, if Γ ∗ D admits a Γ-invariant filtration Li where Li /Li−1
is Γ-generated by a ki -dimensional subspace Di ⊂ Li /Li−1 for all i = 1, 2, ...,
then
X
lim sup |Hj ∗ D|/|Hj | ≤
ki .
j→∞
i
Furthermore, if the group Γ is infinite, while |D| < ∞, then the filtration F
obtained by taking the Γ-span of a filtration 0 = D0 ⊂ D1 ⊂ D2 ⊂ ... ⊂ D with
|Di /Di−1 | = 1, satisfies
|F∂Γ (D)| ≥ lim sup |Hj ∗ D|/|Hj |.
j→∞
All this motivates the definition of the asymptotic rank (dimension)
|D : Γ|H = |D : Γ|{Hj } = lim sup |Dj |/|Hj |,
j→∞
where |D| = rank(D) and |Hj | = card(Hj ). Then one defines
|D : Γ| = sup |D : Γ|H
H
P∗sup (ni ; Γi )
and P∗ (ni ; Γi ) =
with Γi ⊂ Γ as in the case of P◦ , but now with
|D : Γi | instead of |Y /Γi |.
Remark. Instead of ”sup lim sup” one may use other ”generalized limits”
that give non-ambiguous definitions , see [6], [7], [14]. For example, one may
consider lim inf |H|→∞ |H ∗ D|/|H| over all finite subsets H ⊂ Γ, where this leads
to a meaningful (albeit not apparently useful) definition of |D : Γ| for all not
necessarily amenable groups Γ.
If H ⊂ Γ is connected in the G-Cayley graph Γ for a generating set G ⊂ Γ,
then |H ∗ D| ≤ |∂|max
G (D)| · |H|; therefore,
P∗ (ni ; Γi ) ≤ F∗ (ni ; Γi , Gi )
for infinite Γi and arbitrary finite generating subsets Gi ⊂ Γi .
Examples. (a) If Γ = Zk , then (obviously) |D : Γ| equals the dimension of
the linear span of D over the field of rational functions in k variables that is the
field of fractions of the group ring of Zk .
P (b) If the action of Γ on L is free i.e. no non-trivial finite linear combination
γ cγ (a) vanishes, e.g. if L equals A, the group algebra of Γ over some field,
where A has no divisors of zero, then, obviously, |Di | ≥ |Hi | and thus |D : Γ| ≥ 1
for all non-zero D ⊂ A.
16
l2 -rank and P∗l2 . If L equals the Hilbert space of l2 -functions on an X with a
free action of Γ, then an alternative to the asymptotic rank is furnished by the
von Neumann l2 -dimension |D : Γ|l2 defined for all closed Γ-invariant subspaces;
if D ⊂ L is not Γ-invariant, the notion of l2 -dimension applies to the Γ-span
Γ ∗ D of D that is the minimal closed Γ-invariant subspace containing D,
|D : Γ| = |D : Γ|l2 =def |Γ ∗ D : Γ|l2 .
Clearly, |D : Γ|l2 ≤ |D : Γ|{Hj } , whenever the two ranks are simultaneously
defined. Furthermore, if D has a finite support in X, then (it is also well known
and obvious) |D : Γ|l2 = |D : Γ|{Hj } for an arbitrary Følner exhaustion Hj .
(Thus, one does not have to worry here about the existence of the limit in the
definition of |D : Γi |)
If one tries to straightforwardly define the P∗ = P∗l2 -function with |D : Γ|l2 ,
one gets it identically zero already for the coordinate subgroups Γi = Z, i =
1, 2, ..., k, acting on Γ = Zk but, whatever the definition of P∗l2 , it depends on
the Hilbert structure of L, while the above P∗ needs only the linear algebraic
data and P∗l2 ≤ P∗inf whenever the two P -functions are simultaneously defined.
Furthermore, if the space D of functions on X in the definition of P∗l2 has
finite support, then P∗l2 = P∗inf = P∗sup , where ”inf ” and ”sup” refer to the
”inf lim inf” and ”sup lim sup” definitions of the asymptotic rank.
Comparison between the Graph Metrics and the Lattices of Subsets and of
Linear Subspaces. The lattice of the finite subsets Y in a set X is fully determined by the marked edge graph of the X-cube Z = 2X of all finite subsets
in X, where the vertices corresponds to the finite subsets Y ⊂ X and where
one marks the vertex corresponding to the empty set. For example the distance
between two subsets equals one half of the minimal edge length of the paths
between the corresponding vertices.
Similarly, the lattice of finite dimensional linear subspaces D in a linear
space L is determined by the 1-skeleton of the corresponding spherical building
that is the graph, where the vertices correspond to the finite dimensional linear
subspaces D ⊂ X, where the two vertices D1 and D2 are joined by an edge if
and only if one of the two subspaces contains the other with codimension 1, and
where one marks the vertex corresponding to the 0-space.
The evaluation of P∗ (ni ) as well as of the isoperimetric profile I∗ (r; Γ, G) may
be more difficult than that of their combinatorial ◦-counterparts (essentially)
because the lattice of linear subspaces is not distributive: the intersection of a
D with span(D1 , D2 ) can be strictly greater then the span of the intersections
D ∩ D1 and D ∩ D2 (while Y ∩ (Y1 ∪ Y2 ) = (Y ∩ Y1 ) ∪ (Y ∩ Y2 ) for all subsets
of an X).
1.12
Summary of Results.
1. We shall prove in 2.2, generalizing Elek’s l2 -theorem, that
the action of every non-amenable group Γ on the space lp (Γ) of p-summable
real functions is I∗ -linear for all 1 ≤ p < ∞.
We define in 2.3 a class of groups Γ that includes the free groups Fk for
k ≥ 2 and many (but not all) non-amenable groups that contain no F2 , such
that their group algebras are F-non-amenable (or IF -linear) in the following
sense.
17
There exists a finite family of subgroups Γi ⊂ Γ in each group Γ from this
class, such that every linear subspace D in the group algebra of Γ over every
field F has
maxi |F∂Γi (D)| ≥ ε|D|
for some ε = ε(Γ) > 0.
2. We show in 3.2 (elaborating upon a remark by Dima Grigorev who
pointed out to me how the polynomial Brunn-Minkowski inequality reduces to
the combinatorial one with an order on the set of monomials) that
If Γ admits a left invariant order, then the linear algebraic profile equals the
combinatorial one,
I∗ (r; Γ) = I◦ (r; Γ)
and prove some related results.
3. We combine in 6.2 the Coulon-Saloff-Coste argument with the l2 -LoomisWhitney inequality and prove, for example, that
the Følner function F∗ (n) of the Hilbert space l2 (Γ) of square summable
functions on every polycyclic non-virtually nilpotent group Γ has exponential
growth. (For the spaces of functions with finite supports this follows directly
from the Coulhon-Saloff-Coste-inequality and 2).
4. We shall apply the entropic formalism (that mimics the martingale
method in isoperimetry) to the group algebras of normal extensions and thus,
prove, for example, that
Grigorchuk groups (that may have subexponential growth G◦ ≺ exp(nα ) for
0 < α < 1 and may be pure torsion) satisfy
F∗ (n) exp(nβ ) for some β > 0,
(see 7.1,7.2).
Also we prove in 7.2 that
the (twice or more) iterated wreath products Γ of infinite groups, e.g. of
(pure torsion amenable) Aleshin-Grigorchuk groups, have
F∗ (n) exp(exp(n)).
5. We shall exhibit in 8.1, 8.2 a class of amenable groups Γ, where
the combinatorial Følner functions F◦ (n; Γ) may grow arbitrarily fast
(these Γ are extensions of locally finite groups as in the last remark in section
3 of [9]) and, yet,
all these Γ have bounded linear algebraic profiles,
I∗ (r; Γ) ≤ const = const(Γ).
On the other hand we produce
orderable amenable groups Γ where both, combinatorial and linear algebraic
Følner functions, grow arbitrarily fast.
(These Γ are extensions of locally nilpotent groups, rather than of locally
finite groups that were suggested in [9]. Possibly, one can also render orderable
18
the construction of groups with subexponential growth from [10] and make not
only F◦ , as it is done in [10], but also F∗ grow arbitrarily fast.)
6. We present in 9.1-9.3 a translation of the combinatorial argument from
[9] to the language of partitions that improves the lower bounds from [9] on
Følner functions of the iterated wreath products and related classes of groups.
7. We shall prove in 10.1-10.3 some Brunn-Minkovski type inequalities. for
discrete groups and their group algebras.
Our presentation is self contained and a significant part of the paper is
expository.
Acknowledgment. What follows is an outgrowth of a section in my (still
unfinished) paper [16] turned into a separate article by a suggestion by Slava
Grigorchuk. A part of this paper was written during my visit to the mathematics department of the Northwestern University in a stimulating and hospitable
atmosphere. Also I want to thank Tullio Ceccherini-Silberstein, Christophe Pittet and the anonymous referees who generously spent their time in reading the
paper, pointed out a variety of errors and made many useful suggestions.
2
2.1
Non-Amenability of Groups and Strengthened
non-Amenability of Group Algebras.
l2 -Non-Amenability of the Complex Group Algebras
of non-Amenable groups.
Elek non-Amenability Theorem. If Γ is non-amenable, then the isoperimetric profile of the group algebra of Γ over F = C, and, hence over any field
of characteristic zero, is linear:
rank((G ∗ D)) − rank(D) ≥ ε · rank(D)
. for all G ⊂ Γ generating Γ, all finite dimensional linear subspaces D ⊂ A and
some constant ε = ε(Γ, G) > 0.
This is shown in [6] in the following stronger form.
l2 -Theorem. Let Γ be a non-amenable group and G ⊂ Γ be a finite generating set. Then there exists a positive constant ε such that for every N dimensional linear subspace D in the complex Hilbert space l2 (Γ) there exists
γ ∈ G such that dim(γ(D) ∩ D) ≤ (1 − ε)dim(D), where γ(D) refers to the left
action of Γ on l2 (Γ).
Proof. If Γ is non-amenable, then, according to one of the definitions of
(non)amenability,
there is no almost invariant probability measure on Γ: every real positive
l1 -function f on Γ satisfies
maxγ∈G k γ(f ) − f kl1 ≥k ε0 f (γ) kl1
for some positive ε0 depending on Γ and G .
Denote by eγ ∈ l2 = l2 (Γ) for γ ∈ Γ the (delta) function that equals 1 at γ
and vanishes at all other points in Γ and let fD = fD (γ) denote the square of
the l2 -norm of the orthogonal projection of eγ to D regarded as a function on
Γ.
19
Since eγ ∈ l2 = l2 (Γ) for all γ ∈ Γ make an orthonormal basis in l2 , one
has k fD kl1 = dim(D) for all finite dimensional D ∈ l2 and k fD − fD0 kl1 ≤
dim((D+D0 )/(D∩D0 )). In fact, if a subspace D+ ⊂ l2 splits into the orthogonal
sum of D1 and D2 then fD+ = fD1 + fD2 by Pythagorean theorem.
Therefore, k γ(fD ) − fD kl1 ≤ 2dim((D + γ(D))/D)) for all γ ∈ Γ and the
proof follows.
2.2
Geometric Proof that Applies to lp (Γ).
Given a finite dimensional Banach space D we denote by µD the normalized
measure on the unit sphere SD of D induced from the Lebesgue measure on D:
the measure of a subset U ⊂ SD equals the measure of the cone over U in the
unit ball in D divided by the Lebesgue measure of this ball.
Given two finite dimensional subspaces in a Banach space (L, ||...||), say
D ⊂ L and D0 ⊂ L, we denote by µ and µ0 the push-forwards of the measures
µD and µD0 to the unit sphere SL ⊃ SD , SD0 and consider all Borel maps ξ from
SD ⊂ SL to SD0 ⊂ SL sending µ to µ0 .
Define the distance between the two measures by
Z
δ(µ, µ0 ) = infξ
||s − ξ(s)||dµ(s).
SD
Say that L is asymptotically M(easure)D(imension)-consistent if for every
two sequences of finite dimensional subspaces with dimensions converging to ∞
for i → ∞, say Ei ⊂ L and Ei0 ⊂ E, the convergence dim(Ei )/dim(Ei ∩ Ei0 ) → 1
for i → ∞, implies the convergence δ(µi , µ0i ) → 0.
This consistency is classically known for l2 (by the Maxwell distribution law
for the ideal gas) and equally obvious for all uniformly convex Banach spaces,
such as lp for 1 < p < ∞, that these spaces are MD-consistent. (See [18], for
instance).
If an isometric group action on an asymptotically MD-consistent space L has
an almost fixed point on the Grassmannian G ∗ (L) with the normalized metric
|... − ...| (see 1.5) then there exists an almost invariant probability measure on
the unit sphere S in L for the metric δ.
Another relevant property of lp (Γ), is the existence of a Γ-equivariant Lipschitz map (playing the role of the domination map in 3.1,3.2) from the unit
sphere Slp ⊂ l2 (Γ) to the space S1+ ⊂ l1 of the probability measures on Γ: each
lp -function f (γ) goes to f p (γ), where this map (a radial projection in the logarithmic coordinates, compare (c) in 5.4) is invariant under all permutations of
the basic vectors.
Thus every almost invariant measure µ on Slp gives us such a measure µ∗
on the positive ”quadrant” Sl+1 ⊂ Sl1 , that is the space of probability measures
on Γ. Since Sl+1 is a closed convex subset in l1 (Γ) ⊃ Sl1 , each measure on Sl+1
has a center of mass and this center, say µ∗ ∈ Sl+1 , is a probability measure on
Γ itself. Clearly, µ∗ is almost invariant for the l1 -metric whenever the measure
is µ almost invariant for δ . QED.
Remark. The first (Pythagorean) proof of Elek’s theorem, shows that there
is no Hilbert-Schmidt operator on l2 (Γ) that almost commutes with the action
of Γ, while the geometric argument establishes the ”no almost fixed point”
20
property on the Grassmannian of finite dimensional linear subspaces with the
metric induced by the (measure transportation) Monge-Kantorovich metric.
Questions. Let X be a countable set, L some linear space of F-valued functions on X invariant under the group Γ of transformations of X (e.g. the group
of all bijective transformations) and let Grd be the space of all d dimensional
subspaces D ⊂ L with the metric |D1 − D2 | =def d1 |D1 /(D1 ∩ D2 |).
What is the infimum of the Lipschitz constants of the Γ-equivariant maps
from Grd to the space P(X) of the probability measures on X for specific
“natural” spaces L?
What are the Lipschitz constants for equivariant maps between the Grassmannians Grd associated to different spaces L (with, possibly, different F) of
functions on X?
Such maps to P(X) play the key role in the proof of the I∗ -linearity theorem
by Bartholdi; we shall present a version of his construction in 3.6.
Two Comments Made by an Anonymous Referee. (1) It might be interesting to (further, M.G.) geometrize the study of linear isoperimetric profiles
á la Voisculescu, so that one considers not the linear span of translates of a
finite dimensional subspace but rather the smallest dimension of a subspace approximately containing all of these translates granted that the ambient space
is equipeed with a norm. This would seem to tie in with Bekka’s notion of
amenability for unitary representations and it would provide a useful analytic
tool for the study of isoperimetry in spaces other than l2 and group algebras
while at the same time directly connecting to the linear theory in these special
cases, which are the primary objects of study in this paper.
(2) The approximate invariance of a Følner set can be expressed in a dual
way via approximately even coverings with a certain multiplicity. This leads to a
recursive extraction procedure to the quasitiling theorem of Ornstain and Weiss
according to which the Følner set can be approximately tiled by translates of
finitely many tiles at different scales. Might the type of isoperometry technology
elaborated upon in the paper be used, via some duality principle, to study the
problem of the number of different scales required to quasitile Følner sets in
variuos groups (about which little seems to be known)
2.3
F- Non-Amenability of the Group Algebras of Groups
with Acyclic Families of Subgroups.
(A) Topological Argument for Non-amenability of Fk . The free Group
Γ = Fk on k ≥ 2 generators freely acts on the infinite regular 2k-valent tree
Tk . Every finite subtree with the vertex set Y ⊂ X consisting of r vertices yi ,
i = 1, 2, ..., r, of degrees di satisfies, by the Euler-Poincare formula,
r−
1X
di = 1.
2 i
Hence,
|∂ joint (Y )| =
X
(2k − di ) = 2rk − 2r + 2
i
and
I◦ (r; Γ) = r(2k − 2) + 2.
21
It follows that Fk for k ≥ 2 is non-amenable and, by the extension property,
that the groups containing an F2 as a subgroup are non-amenable.
We shall return to the topological aspects of isoperimetry in [16] but now
let us indicate another approach for estimating I◦ that applies, for example, to
Burnside groups.
(B) Slicing Argument. Look at how the orbits (cosets) of the k generating
cyclic subgroups Γi in Γ = Fk meet and ”slice” a finite subset Y0 ⊂ Γ, say for
Γ = F2 .
Since every finite (sub)tree contains at least one leaf (end-vertex) y, there
exists, a subgroup, among the generating two, such that some of its orbits
(cosets) in Γ meets Y0 only at y. (In general, there are k − 1 of such subgroups
at every leaf) . We remove from Y the intersection S1 = S1 (Y0 ) of such an orbit
with Y0 , then remove a similar S2 = S2 (Y1 ) from Y1 = Y0 \ S1 , etc., until we
exhaust all of Y0 . Thus we obtain |Y0 | orbits non-trivially meeting Y0 ; therefore
there are at least |Y0 |/2 orbits of one of these subgroups, say of Λ ⊂ Γ, that
meet Y0 and
|∂Λ (Y )| ≥
1
|Y |;
2
thus the max-boundary of Y in Γ is also bounded from below by 12 |Y |.
(More generally, if k ≥ 2, then |∂|max (Y ) ≥ k−1
k |Y | by the slicing argument.)
This may seem no better than the above bound for |∂ joint (Y )| but the slicing
argument itself carries more power. For example, it implies
(C) F and P∗ -non-Amenabilty of the Group Algebra of Γ = F2 .
Let D0 be a linear space of functions on Γ with finite support Y0 = supp(D0 ).
Then the restriction homomorphism h1 from D0 to (the space of linear functions
on) the above slice S0 ⊂ Y0 clearly has rank 1.
Let D1 = ker(h1 ) ⊂ D0 and let h2 be a similar homomorphism from D1 to
such a slice S2 ⊂ Y1 = supp(D1 ), etc. Thus we obtain, for one of these two
cyclic subgroups, say for Λ ⊂ Γ,
an increasing chain of Λ-invariant subsets, Z0 ⊂ Z1 ⊂ ... ⊂ Zn for n ≥
|D0 |/2, such that the kernels Di0 ⊂ D0 of the restriction homomorphisms from
D0 to (the spaces of functions on ) Zi0 strictly(!) decrease with i.
It follows that the filtered boundary and the asymptotic rank (see 1.11)
satisfy
|D : Λ| ≥ |F∂Λ (D)| ≥ n ≥ |D0 |/2.
Indeed, take a splitting D0 = ⊕i ∆i compatible with the filtration Di0 of
D0 and observe that the spans span(Λ(∆i )) are linearly independent, since
supp(Λ(D)) = Λ(supp(D)) for all linear spaces D of functions on Γ.
This implies the required linear lower bound on |F∂Λ (D)| and thus, such a
bound on the asymptotic rank |D : Λ| that is the dimension of the span of D
over the field of fractions of the (commutative!) group algebra of Λ.
More generally, one has
(D) F-non-Amenability in the Presence of Divergent Families of
Subgroups. Let X be a Γ-space and let Γi ⊂ Γ i = 1, 2, ..., k < ∞ be a 1divergent family of subgroups, which means, by definition, that for every finite
subset Y ⊂ X there is an orbit S ⊂ X with |S| > k of some Γi which satisfies
22
|S ∩ Y | = 1.
Then the action of Γ on the space of F-valued functions on X with finite supports
is IF -linear for all fields F.
Proof. There is an i such that by successively making the function from D
vanish on Γi -orbits Sm we obtain a strictly descending chain Dm ⊂ D of length
≥ |D|/k, where there are at least |D|/k 2 subspaces Dm with |Dm−1 /Dm | <
|Sm |. Since every Dm−1 contains a function supported on a single point in Sm ,
the corresponding quotients |Dm−1 /Dm | are not Γi -invariant.
(E) Corollary: Acyclicity ⇒ IF -linearity. Let Γ admit an infinite sequence of Γj , j = 1, 2, ...,, with cardinalities |Γj | > k, where k equals the number
of different subgroups among Γj , such that
γj1 · γj1 +1 · ... · γj2 −1 · γj2 6= id
for id 6= γj ∈ Γj and all j1 ≤ j2 . (Such sequences are called acyclic.) Then the
group algebra of Γ over any field is IF -linear.
Proof. If a subset Y ⊂ Γ has |Y ∩ S| ≥ 2 for all orbits (cosets) of all Γj that
meet Y , then Y contains infinitely many γ of the form γ0 , γj0 ·γj1 , γj0 ·γj1 ·γj1 +1 ,...
for γ0 ∈ Y and γj ∈ Γj for j ≥ 1.
Examples. (a) If Γ equals a free product, Γ = Γ1 ∗ Γ2 , then the sequence
Γ1 , Γ2 , Γ1 , Γ2 , ... is acyclic and the above applies to infinite Γ1 and Γ2 . ( This
argument yields the non-amenabilty of Γ itself for |Γ1 | ≥ 2 and |Γ2 | ≥ 3 as
everybody knows).
(b). Consider a square free sequence ij ∈ I = {1, 2, ..., k ≥ 3}, i.e. where
there are no subsequences ij , ij+1 , ..., ij+2l , where ij+s = ij+l+s for s = 0, 1, ...l.
Then take a free product Γ0 = Γ1 ∗Γ2 ∗...∗Γk , enumerate all γ ∈ Γ by m = 1, 2, ...,
nm
and add to Γ torsion relations γm
= id one after another, omitting those where
the corresponding γm was already torsion in the preceding factor group.
The hyperbolic small cancellation tells us that the groups Γi inject into the
resulting factor group Γ for torsionless Γi and fast growing nm and that the
sequence of subgroups Γij is acyclic in the above sense for square free sequences
ij .
Moreover, (this was explained to me by Azer Akhmedov, Ashot Minasyan
and Dima Sonkin) this property is known for free Burnside p-torsion groups Γi
and all nm = p for large enough p.
The Burnside groups B(p, 2) on 2 generators contain B(p, k) for all k = 6;
thus, they contain the p-torsion quotients of B(p, 2) ∗ B(p, 2) ∗ B(p, 2); therefore,
(F) The groups B(p, k) are IF -linear for all sufficiently large p and k ≥ 2.
Remarks. (a) Random quotients Γ of all resilient (see [15]), e.g. of Burnside,
groups are not acyclic in the above sense (where some of them are ”Olshanskiy
monsters” with all their proper subgroups isomorphic to Zp , as was pointed out
to me by Dima Sonkin, but being a monster does not by itself exclude acyclicity)
and it is unclear which of them possess some ”strengthened non-amenability”
controlled by the combinatorics of orbit partitions and/or of invariant filtrations
in the linear algebraic setting. (The ordinary non-amenability of such groups
follows by the Grigorchuk criterion and also can be derived from Kazhdan’s
T -property in some cases.)
23
(b) We emphasized the IF -linearity(= F-non-amenability) rather than the
(weaker property of) I∗ -linearity(=linear algebraic non-amenability), since the
latter is valid by the Elek-Bartholdi theorem for all non-amenable groups Γ.
3
Isoperimetry in the Group Algebras of Ordered Groups.
We show in this section that the isoperimetric inequalities in the (left) orderable
groups Γ pass from subsets in Γ to linear subspaces in the group algebra of Γ
over any field.
3.1
Triangular Systems and Bases.
Let X = (X, ) be an ordered set, e.g. X = {1, 2, ..., n}, take a function f on X
with values in some field (any set with a distinguished element called 0 will do
for the following definition) and denote by xmin
∈ X the minimal x ∈ X where
f
, that the
f (x) 6= 0, where we assume, to guarantee the existence of such xmin
f
function f has a finite support S = Sf ⊂ X or, at least Sf is bounded from
below, i.e. Sf contains at most finitely many elements that are x0 for every
x0 ∈ X. We also agree that xmin
f =0 equals the empty set.
Given a linear space D of functions on X with the supports uniformly
bounded from below, assign to it the union of the points xmin
for all f ∈ D,
f
denoted
[
{xmin
}⊂X
D 7→ YDmin =
f
f ∈D
and observe that
the cardinality of YDmin equals the rank of D,
|YDmin | = |D|.
Example. If X = {1, 2, ..., n} then YDmin equals the set of the non-zero
diagonal entries in a triangular matrix with n columns and |D| rows where the
rows make a basis in D.
3.2
Domination by Ordering and Derivation of the Linear
Algebraic Isoperimetric Inequality from the Combinatorial One.
Isoperimetric Domination. Say that the isoperimetry of a linear space L acted
upon by a group Γ dominates the isoperimetry of some set X (non-linearly)
acted by Γ, if there exits a Γ-equivariant (domination) map from the set G ∗
of finite dimensional subspaces D ⊂ L to the set 2X of subsets Y ⊂ X, say
D 7→ YD , such that
|YD | = |D| for all D
and
24
(∗ ◦)
|span(Di )| ≥ |∪i YDi |
for all finite collections of subspaces Di .
Clearly,
the map D 7→ YDmin based on a Γ-invariant order in X is an isoperimetric
domination.
It is also clear that
the existence of an isoperimetric domination makes I∗ (r; L, Γ) ≥ I◦ (r; X, Γ).
This leads to the following
Equivalence ◦ ⇔ ∗ with Order. Let an action of Γ on X preserve some
order on X. Then the linear algebraic isoperimetric profile I∗ of (the space of
functions with finite supports on) X with respect to an arbitrary subset G ⊂ Γ
majorizes the combinatorial profile I◦ of X, and since, obviously, I∗ ≤ I◦ , the
two profiles are equal,
(◦ ⇔ ∗)
I∗ (r; X, Γ) = I◦ (r; X, Γ).
Futhermore, the P -functions of X with respect to (the family of the orbit
partitions of) arbitrary groups Γi acting on X satisfy
P∗ (ni ) = P◦ (ni ).
Remark.(a) Every domination map G ∗ → 2X is distance decreasing for the
Γ-invariant metrics in G ∗ and in 2X defined in 1.1, 1.4 and 1.5; this serves as
good as the inequality |span(Di )| ≥ | ∪i YDi | if we work with the I max -profiles,
(compare lp -non amenability in 2.2), while the profiles defined with the joint
boundary (see 1.1) need more of the (lattice) structure in the spaces G ∗ and in
2X .
3.3
Examples of Orderable Groups and Corollaries to The
Equivalence ◦ ⇔ ∗.
Left orderable groups constitute an ample class of groups starting with Γ = Z.
They are built with the following constructions.
(1) Extensions. An extension Γ of a left orderable group Γ0 by a left orderable
Γ1 is left orderable by the lexicographic ordering construction. In particular
torsionless polycyclic groups are orderable
and thus their isoperimetry extends to their group algebras.
(2) Limits and Subgroups. Inductive and projective limits of orderable groups
are, obviously, orderable and subgroups of orderable groups are orderable. It
follows that
If every non-trivial finitely generated subgroup in Γ admits a surjective homomorphism to Z then Γ is left orderable.
In particular, the following groups are orderable,
(a) free groups Fk , (b) wreath products Γ of orderable groups, (c) Z-extension
of torsion free locally nilpotent groups by Z.
Thus the Equivalence ◦ ⇔ ∗ with order gives us yet another proof of the
Elek theorem on non-amenability of the group algebras of Fk for k ≥ 2, and
shows together with 8.2 that
25
the group algebra over a given field of a finitely generated amenable group
may have arbitrarily fast growing Følner function.
(3) Finite groups, and hence, groups with non-trivial torsion, admit no left
invariant order. In fact, if a linear space D consists of functions on Γ that are
constant on the cosets of a finite subgroup Γ0 ⊂ Γ and G ⊃ Γ0 , then, obviously,
I∗ (|D|) < I◦ (|D|).
(4) Amenable groups are orderable if and only if all their non-trivial (i.e. 6=
{id}) infinite subgroups admit epimorphisms to Z [22]. The ”if” part mentioned
earlier is valid for all groups and is rather obvious.)
Thus, many virtually polycyclic (e.g. virtually free Abelian) groups are not
orderable; Yet,
if Γ is commensurable to a left orderable group, then I∗ is equivalent to I◦
for Γ.
In fact, if Γ1 ⊂ Γ has finite index, or if Γ admits an epimorphism on Γ1 with
a finite kernel , then the isoperimetric profiles of the two groups as well as of
their group algebras are equivalent.
(5) The group homeo+ (R) of orientation preserving homeomorphisms of the
real line is left orderable and every countable left orderable group embeds into
homeo+ (R). (This is well known and easy.) For example, the universal covering
of SL2 (R) is left orderable and also
the (finitely generated) Thompson group T (dyadic homeomorphisms of an
interval) is left orderable; thus, the (yet unknown) profile I◦ of T is equal to I∗
of T .
3.4
I∗ -Profiles of Groups with Orderable Subgroups.
Let a finite subset G0 ⊂ Γ generate an orderable group Γ0 ⊂ Γ, where G0 ⊂ G·m
for a finite generating subset G ⊂ Γ. Then the inequality G·m from 1.4 combined
with the equivalence ◦ ⇔ ∗ shows that
(∗)
I∗max (r; Γ, G) ≥
1 max
I
(r; Γ0 , G0 ).
m ◦
Examples.(a) Observe that the relative growth of Γ0 ⊂ Γ, denoted G◦◦ (n) =
|G·n ∩ Γ) | may be faster than the growth of Γ0 with respect to any finite generating subset in Γ0 if Γ0 is “sufficiently distorted” in Γ. Then the following
inequality, that derived from the above along with the Displacement Inequality
( see 1.4), may serve better than just F∗ (n, Γ) ≥ F∗ (n, Γ0 ),
n
1
G◦◦ ( ).
2
2
However, I see no example, with a finitely generated Γ0 , where one can not do
better by other means.
(b) If Γ0 is infinitely generated, e.g, being the restricted Cartesian power of
some finitely generated group Γ1 , then one can use measures µ on Γ0 spread over
large sets of “independent” elements in Γ0 . Such situation arises, for example for
the wreath products Γ of orderable finitely generated groups Γ1 with finitely generated (not necessarily orderable) Γ2 , where one can apply the Loomis-Whitney
inequality and obtain the lower bound
F∗max (n; Γ, G) ≥
26
F∗ (n; Γ) exp(G◦ (n; Γ2 ).
But, in truth, this holds for all (not necessarily orderable) groups Γ1 (see 7.2)
and if Γ1 is not pure torsion, such an inequality is also valid for real l2 -functions
and for decaying functions on Γ with values in ultra metric fields (see 6.2).
3.5
I∗ -Profiles of Groups with Orderable Factor Groups.
Consider an exact sequence,
1 → Γ1 → Γ → Γ2 → 1
and let Gi ⊂ Γi , i = 1, 2, be finite symmetric generating subsets in Γi containing
the identity elements, where G2 also denotes some lift of G2 to Γ.
Orderable Extension Inequalities. Let Γ2 be a left orderable group.
Then the isoperimetric profile of the group algebra A of Γ is bounded from below
by that of Γ2 ,
I∗ (r; Γ, G) ≥ I∗ (r; Γ2 , G2 ) = I◦ (r; Γ2 , G2 )
for G = G1 ∪ G2 ⊂ Γ.
Proof. Denote by h the homomorphism Γ → Γ2 , let Aγ2 ⊂ A, where γ2 ∈
Γ2 , consist of the functions a on Γ such that all γ in the support supp(a) ⊂ Γ
satisfy h(γ) γ2 for a given left order on Γ2 and define A≺γ2 ⊂ Aγ2 with the
strict inequality h(γ) ≺ γ2 .
Given a linear subspace D ⊂ A, let
r = r(γ2 ) =def rank(D ∩ Aγ2 )/(D ∩ A≺γ2 )
and observe that
The G2 -boundary of the function r on Γ2 (i.e. the boundary of the subgraph
of r in Γ2 × Z+ ) is bounded by the G2 -boundary of D,
|∂G2 (r)| ≤ |∂G2 (D)|;
and the bound on I∗ (Γ) by I◦ (Γ2 ) follows as in 3.2.
Remark. We shall isolate in 5.11 a class of •-convex functions F and prove
that
F∗ (n1 , n2 ; Γ, G1 , G2 )) ≥ F∗ (n1 ; Γi , G1 ) · F◦ (n2 ; Γ2 , G2 ),
when the functions F∗ (n1 ; Γi , G1 ) ·F◦ (n2 ; Γ2 , G2 ) are •-convex .
3.6
Proof of Bartholdi’s I∗ -Linearity Theorem via Orderings.
The set O(X) of all orders O on a countable set X equals the projective limit
of the orders on the finite subsets Y in X; thus, it carries the structure of a
compact topological space. The natural action of the group Γ of all bijective
transformations of X is continuous on O(X), where it is free and transitive for
finite X. (If X is a finite set, then O(X) equals the set of the top dimensional
simplices in the first barycentric subdivision of the regular n-simplex for n =
27
|X| − 1.) It is non-transitive for infinite X as there are non-isomorphic orders
and it is not free: the isotropy subgroup of a point O in O(X) consists of the
transformations preserving the order O.
There is a unique Γ-invariant probability measure, say dO on O(X), that
is the projective limit of the uniform measures on O(Y ) for the finite subsets
Y ⊂ X.
Given an order O we invoke the (triangular basis) map from the set of all
finite dimensional linear spaces of F-functions on X to the set of finite subsets
subsets in X,
D 7→ Y = YDmin = YDmin (O)
associated to this O, and, following [2], let µY = µO
D be the measure on X which
assigns the unit weights to all points in Y ⊂ X and which is zero outside Y .
Clearly,
(=)
|µO
D | = |D|
and
(≤)
O
D1 ⊂ D2 ⇒ µO
D1 (x) ≤ µD2 (x)
for all x ∈ X.
Then we take the average over all orders O,
Z
µD =def
µO
D dO
O
and observe that the measure µD on X, being an average, inherits the relations
(=) and (≤) from µO
D ; furthermore, since the measure dO is Γ-invariant the map
D 7→ µD (unlike D 7→ µO
D for a fixed order O) is Γ-equivariant.
The relations (=) and (≤) immediately imply (compare [2]) that ||µD1 −
µD2 ||l1 = |D1 | − |D2 | whenever D1 ⊂ D2 and, hence, the map D 7→ µD is 1Lipschitz for the graph metric (see 1.11) in the space of D’s (that is equaivalent
to the metric defined in 1.6) and the l1 -metric in the space of measures; then
the implication non-amenability ⇒ I∗ -linearity follows as in 2.2.
Remarks. (a) The above argument is a minor modification/simplification of
the original one in [2], where the author defines the set MD ⊂ l1 (X) ⊂ l2 (X) of
the measures µY on X associated to the subsets Y ⊂ X, such that the restriction
map D 7→ D|Y is isomorphic, and then averages with the measure ds on MD
which is induced by the (multivalued) Gauss spherical (support) map of MD to
the unit sphere in l2 (X).
(b) The (compact topological) Γ-spaces of orders on countable groups Γ seem
to have interesting dynamics, where, in particular, the presence of ”dynamically
small” invariant subsets and/or measures (e.g. with vanishing Kolmogorov entropy) may yield ”good isoperimetry” in the linear spaces of functions on Γ.
28
4
Families of Measurable Partitions and their
Invariants.
A Borel measurable (not necessarily measure preserving) map between Borel
measure spaces, P : (X, λ) → (X 0 , λ0 ) gives rise to a partition of X into slices
that are the pullbacks of points in X 0 . If the map is surjective, we ascribe
the notation P to this partition, identify X 0 with the quotient space X/P and
denote the slice of P through x ∈ X by P (x) ⊂ X and using the same notation
for P (x) ∈ X/P as well as P −1 (x0 ), x0 = P (x) ∈ X/P .
These slices carry (almost everywhere defined and a.e. unique) Fubini measure, denoted λ/λ0 and also dP (x), such that
Z
Z
Z
f (x)dλ =
dλ0
f (x)dP (x)
X
X/P
P (x)
. for all measurable functions f on X.
Coordinate Lines and Planes Partitions. Every Cartesian product space,
X = ... × Xi × ..., i ∈ I,
comes along with an I-family (”web”) of partitions, each into the i-th ”coordinate lines”, that are the copies of Xi passing through the points in X. (These are
the true parallel coordinate lines for the Euclidean spaces X = R|I| = ...×R×...
with the coordinates indexed by i ∈ I.) Moreover, X supports the ”web” of 2I
partitions corresponding to the “planes” indexed by subsets J ⊂ I and representing ×i∈J Xi .
Induced Partitions. If Y ⊂ X is a measurable subset, then the intersections
of Y with slices of a partition P of X are called P -slices (” threads”) of Y . Thus
we obtain a partition P |Y of Y , where we disregard the empty intersections (or
rather those of measure zero).
Orbit Partitions. If X is acted by a group Γ then it is partitioned into the
orbits of the action, where this partition is measurable for (measurably) proper
actions.
In most our examples we deal with countable, also called discrete, spaces
where all partitions are measurable. If such a space X is unitary, i.e. all atoms
(points) have unit weights, then every partition P equals the orbit partition of
the group of the transformations of X preserving the slices of P .
Remark. The only significant invariant of a measurable partition is the
distribution of the values of the function λ(P (x)) on X (or, equivalently, of the
function λ(P −1 (x0 )) on X 0 = X/P ). In particular, all partitions of discrete
unitary spaces into countably many countable slices are mutually isomorphic.
On the contrary, ”measurable webs” that are families of partitions are vastly
different and, typically, very rigid.
Example. The partitions of the 3-dimensional Heisenberg group H into the
orbits of the three standard generating subgroups uniquely determine the Lie
groups structure in H (this is easy); probably, the same remains true for all
families of orbits partitions of groups except for the obvious “exceptionally soft”
cases (such as Z k generated by k copies of Z).
In the following sections we introduce several invariants of families of partitions starting with the well known picture of the isoperimetry in the free Abelian
groups serving as a motivating example.
29
4.1
Four Proofs of the Isoperimetric Inequality for Zk .
Zk -Inequality. The max-Følner function of the free Abelian group Zk with the
standard k generators equals nk .
The non-trivial part of the claim, the inequality
F◦max (n; Zk ) ≥ nk ,
follows by induction with either (a), (b) or (c) below.
Lemma. Let a countable set X be acted upon by groups Γ1 and Γ2 with
given finite generating subsets and Γ be the group generated by Γ1 and Γ2 .
(a) If the Γ1 - and Γ2 -orbits are transversal, i.e. intersect at single points,
and if the Γ2 -orbits are infinite,
or
(b) if the transformations from Γ2 send Γ1 -orbit to Γ1 -orbits and the orbits
of the resulting action of Γ2 on the quotient space X/Γ1 are all infinite, then
F◦max (n; X; Γ) ≥ n · F◦max (n; X; Γ1 ).
(c) If Γ2 sends Γ1 -orbits to Γ1 -orbits as in (b) and all Γ1 -orbits are infinite,
then
F◦max (n; X; Γ) ≥ n · F◦max (n; X/Γ1 ; Γ2 ).
Proof of (a) . Take a Y ⊂ X and look at the two partitions of it where the
slices are the intersections of Y with the Γ1 - and Γ2 - orbits. If
|∂Γ1 (Y )| ≤
1
|Y |,
n1
then there is a Γ1 -slice of Y , say S1 , such that
|∂Γ1 (S1 )| ≤
1
|S1 |
n1
and by the definition of the Følner function for (X, Γ1 ),
|S1 | ≥ F◦max (n1 ; X, Γ1 ).
Since the Γ2 -orbits are infinite, each non-empty Γ2 -slice of Y contributes at
least one point to the Γ2 -boundary and the number of the Γ2 -slices that meet
S1 is ≥ |S1 | due to the transversality of the two partitions. If, furthermore,
|∂Γ2 (Y )| ≤
1
|Y |,
n2
then
|Y | ≥ n2 · |∂Γ2 (Y )| ≥ n2 · |S1 | ≥ n2 · F∗max (n1 ; X; Γ1 )
and the required bound on F◦max (n; Γ) follows if we take n1 = n2 = n.
Proof of (b). Consider the subsets Y m ⊂ X/Γ1 of those Γ1 -orbits that meet
Y at ≥ m points, where m = 1, 2, ..., |S1 | for the above S1 , observe that
X
|∂Γ2 (Y )| ≥
|∂Γ2 (Y m )| ≥ |S1 |.
m=1,2,...,|S1 |
30
and conclude the proof as in (a).
Proof of (c). Since
|Y | =
X
|Y m |,
m
and
|∂Γ2 (Y )| ≥
X
|∂Γ2 (Y m )|,
m
the cardinality of Y 1 satisfies
|Y 1 | = maxm |Y m | ≥ F◦ (n2 ; X/Γ2 , Γ2 ).
Since each Γ1 -slice of Y contributes a point to the Γ1 -boundary of Y , the
subset Y 1 ⊂ X/Γ1 , that is the set of the Γ1 -slices of Y , is bounded by
|Y 1 | ≤ |∂Γ1 (Y )| ≤
1
|Y |
n1
and the proof follows.
Multivariable Følner Function of Zk . Recall that the multivariable Følner
function F◦ (ni ; X, Gi ) for several sets of transformations Gi acting on X is
defined as the maximal N such that every Y ⊂ X with
|∂Gi (Y )| ≤
1
|Y |
ni
has
|Y | ≥ N.
Multivariable Isoperimetric Inequality for Zk .
F◦ (n1 , ..., nk ; Zk , g1 , ..., gk ) = n1 · ... · nk
for every set of generators g1 , ..., gk in Zk .
Proof. Since the orbits S of all k subgroups Γi = Z in Zk generated by gi are
infinite, every S that meets a finite set Y ⊂ Zk contributes at least one point
to its boundary; thus, |∂gi (Y )| is bounded from below by the number mi of the
Γi -orbits S that meet Y . Q
k−1
These numbers satisfy
by the Loomis-Whitney inequality
Q i mi ≥ |Y |
(see 1.10); hence, F◦ ≥ ni , while the opposite inequality is obvious. QED.
About F∗ . It is unclear how much of (a),(b) and (c) remain valid for linear
actions. Yet, we shall see in 6.2 that (b) holds for Γ acting on the space of
functions on X with finite supports if, for example, there is an element of
infinite order in Γ2 freely acting on X/Γ1 . Also we prove in 7.2 a (coarse
entropic) version of (b) for F∗ in the general case.
4.2
Følner Transform F and Schwartz Symmetrization.
Given a family P of measurable partitions Pi , define the Følner transform (composition) F : (Fi ) 7→ F = F (ni ), where Fi = Fi (n), n ∈ [0, ∞), are monotone
increasing positive functions, as follows. Take Ii = IFi , for IFi (r) = rJi (r),
31
where Ji (r) stands for the inverse function of Fi (n) ( i.e. F = FI for this I compare 1.9), take the integral over Y /Pi of the values of Ii at the Fubini measures
of all Pi -slices S of Y and denote
Z
|∂|Ii (Y ) =
Ii (|S|).
Y /Pi
Finally, let F (ni ) = F (Fi (ni )) be the maximal N , such that every Y with
|∂|Ii (Y ) ≤ n1i |Y | has |Y | ≥ N .
Example. If a (discrete or locally compact) group Γ is partitioned into the
orbits of (closed ) subgroups Γi ⊂ Γ, then the multivariable Følner function of Γ
with respect to given generating subsets Gi ⊂ Γi , obviously, satisfies (with the
obvious extention of the notations to the locally compact non-discrete case).
F◦ (ni ; Γ, Gi ) ≥ F (Fi (ni ; Γi , Gi )).
P◦ -Functions and the Følner Transform . The P◦ -function defined in 1.10
clearly, equals the Følner transform of the functions F (ni ) = ni .
Schwartz Symmetrization. If X = X1 ×X2 with the two coordinate partitions
then the corresponding Følner transform (composition) is denoted
F1 • F2 =def F (F1 , F2 ; X1 × X2 ).
It is obvious that F1 • F2 ≤ F1 · F2 .
Let us model the general product space by X = R2+ , where we consider
closed monotone subsets Y ⊂ X which means that the intersections of Y with
the vertical lines, x1 ×R+ are segments x1 ×[0, x2 = f1 (x1 )] and the intersections
with the horizontal ones, R+ × x2 , are segments [0, x1 = f2 (x1 )] × x2 , where f1
and f2 are mutually inverse monotone decreasing functions with the common
graph serving as the boundary of Y .
Let Ii (r) = IFi (r), for i = 1, 2, and 0 ≤ r < ∞, be ”the isoperimetric
profiles” corresponding to Fi , set
Z ∞
Ii (Y ) =
Ii (fi (r))dr
0
and define F (n1 , n2 ) as the infimum of the measures µ of monotone subsets
Y ⊂ R2+ that satisfy
Ii (Y )/µ(Y ) ≤ ni .
It is easy to see (we shall not use it in sequel) that
if Ii are positive concave monotone increasing functions, then this F equals
the Følner transform of the functions Fi corresponding to Ii ,
F = F1 • F2 .
For example, if the extremal Y = Y (n1 , n2 ) are rectangular for all n1 , n2 , ≥ 0,
( i.e. Y = [0, x1 ] × [0, x2 ] ⊂ R+ ) then the the Følner functions Fi corresponding
to Ii satisfy F1 • F2 = F1 · F2 .
32
In general, the extremal Y are bounded by broken lines consisting of finitely
many vertical and horizontal segments under mild regularity/genericity assumptions on Ii and thus the problem of evaluation F = F1 • F2 = F1 · F2 reduces to
a finite dimensional variational problem.
Similar considerations also apply to the Cartesian products of more than
two spaces. but this does yield any explicit formula for the Følner transform,
nor does this deliver a transparent criterion for the equality F1 • F2 = F1 · F2 .
On the brighter side, we shall prove in 5.11 this equality, under the assumption of convexity of logF ( 1ε ).
Schwartz Symmetrization for Group Extensions.
Let Γ be an extension of Γ1 by Γ2 ,
1 −→ Γ1 −→ Γ −→ Γ2 −→ 1,
let Gi ⊂ Γi , i = 1, 2, be generating subsets and let us, to save notation, use G2
for some lift of G2 to Γ. Then the maximal Følner Functions of Γ for G1 and
every lift G02 ⊂ Γ of G2 to Γ are bounded from below by the Følner functions Fi
of Γi , i = 1, 2 in the same way as the Følner function of the Cartesian product
Γ1 × Γ2 is,
F◦ (Γ) ≥ F1 • F2 .
Proof. Given a subset Y ⊂ Γ denote by m(γ) , γ ∈ Γ2 , the number of
points in Y which go to γ under the homomorphism Γ → Γ2 and consider
the subgraph YS ⊂ Γ2 × Z+ of the function m(γ), that is the set of pairs
(γ ∈ Γ, i = 1, 2, ..., ) such that m(γ) ≤ i. Then the proof follows by observing
that |∂G2 (YS )| ≤ |∂G02 (Y )| for the obvious action of Γ2 on Γ2 × Z+ .
Remark. In order to apply this, we need a computable lower bound on the
•-product. A rough such bound follows with the F E◦ inequality in 4.5 and
the sharp one is established in 5.11 for the entropic Følner Functions F• (n),
where logF• ( 1ε ) is, by definition, convex.
Question. Suppose the ratio
I◦ (r; Γ, G1 , G2 )/I◦ (r; Γ1 × Γ2 , G1 , G2 )
remains bounded as r → ∞. Is then Γ commensurable with Γ1 × Γ2 , provided
the groups are amenable?
(Also the random walk on a non-trivial extension of groups, probably, dissipates faster than that on the products as it happens to the Brownian motion on
vector bundles according to the Kac-Feynman formula that would be, for group
extensions, a comparison relation between the G1 -random walk on Γ1 with the
dissipation rate of the measures on Γ1 ⊂ Γ coming from the paths in Γ ⊃ Γ1
that return to id down in Γ2 .)
4.3
E◦ -Functions and E◦ -inequalities.
Definition of E◦ . Given a measure space X with a countable family of partitions
Pi , i = I, define the (expansion) E◦ -function E◦ (Li ) = E◦ (Li ; Pi ) for Li ∈ [0, ∞]
as the supremum of the functions E(Li ), such that every subset Y ⊂ X whose all
(non-empty!) Pi -slices have their (Fubini) measures ≥ Li for all i ∈ I, satisfies
the following lower bound on its own measure,
33
|Y | ≥ E◦ (Li ).
(Compare Lemma 1 in [9].)
Example: E◦ for Cartesian Products. Let X be the Cartesian product,
X = ×i∈I Xi with the product measure, and let Pi be the coordinate “line”
partitions with the respective slices isomorphic to (the copies of) Xi in X.
Then, obviously,
(EΠ)
E◦ (Li ) =
Y
Li .
i∈I
Let us generalize this to the coordinate ”plane“ partitions with the slices
(isomorphic to the copies of ) ×i∈J Xi for (all) subsets J ∈ I.
Define a partition of unityP
on I as an assignment of a non-negative number
αJ to each J ⊂ I, such that
αJ χJ = 1, where χJ : I → {0, 1} ⊂ R denote
the characteristic (indicator) functions of the subsets J ⊂ I.
Coordinate E◦ -Inequalities . The E◦ -function of the family of the partitions PJ is bounded from below by,
(EΠα )
E◦ (LJ ) ≥
Y
J
Lα
J
J⊂I
for all partitions of unity of I..
Proof. The equality (EΠ) shows that the minima l(J) of the logs of the
measures of PJ -slices of Y are sub-additive
l(J1 ) + l(J2 ) ≤ l(J1 ∪ J2 )
for all pairs of disjoint subsets J1 , J2 ⊂ I. Then (EΠα ) follows by the inclusionexclusion principle for sub-additive set functions.
Questions. Can one evaluate E◦ for standard groups (Γ, G) with families of
cyclic subgroups Cg ⊂ Γ generated by g ∈ G? For example, what is E◦ (Γ, Cg )
for the free nilpotent and solvable group Γ on a given generating set G with a
prescribed nilpotency (solvability) degree? Does the function E◦ change much
if one augments the collection Cg with extra (cyclic) subgroups?
Let F be a finite field, Γ be the special linear group and let Γi be some
”standard” subgroups, e.g. the k subgroups SLk−1 (F) positioned inside SLk (F)
in the k obvious ways.
What is the expansion function for these? Can one prove the isoperimetric
inequality for Γ = SLk (F) by some slicing argument? (This would bring us
closer to the T -property for SLk (R).)
Can one meaningfully bound from below the cardinality of G·m ⊂ SLk (F),
say for m = 2, for subsets G ⊂ SLk (F) with |G| ≥ ε · |SLk (F)|, where ε is
1
somewhat greater than 2k
?
4.4
Slice Removal and Slice Decomposition.
Decomposition Lemma. Let Y ⊂ X have the measure < E◦ (Li ), i ∈ I
for given partitions Pi of X and numbers Li . Then Y can be decomposed into
34
disjoint measurable subsets Yi ⊂ Y , such that all Pi -slices of Yi have measures
≤ Li for all i ∈ I.
Proof. Keep removing slices Si from Y of measures ≤ Li .
Corollary: Slicing Inequality for FΣincr .
Let Fi (n) and Ii (r) be some “Følner” functions and the corresponding “profiles”, and let the “boundary” of an Y ⊂ X be understood as the sum of the
Ii -“boundaries”
X
|∂|Σ (Y ) =
|∂|II (Y ).
i
If the functions Ii (r) are monotone increasing, then the “Følner” function“
of (X, Pi ) associated to |∂|Σ (a kind of a diagonal Følner transform of Fi ) satisfies
(FΣ )
FΣ (n; X, Pi ) ≥ Fincr (n) ≥ E◦ (Li = Fi (n)),
where Fincr (n) is a function for which the corresponding I(r) is monotone increasing.
Proof. Decompose a given Y ⊂ X into the union of Yi and observe that
|∂|Ii (Yi ) ≤ |∂|Ii (Y ) by the monotonicity of Ii . Adding these inequalities and
recalling the definitions of Fi and Ii we obtain the required lower bound on
|∂|Σ (Y ).
Remark. This inequality is not sharp. It can be slightly improved by taking
into account, by how much the measure of Y diminishes as we remove subslices
from it; but this makes the inequality only marginally better.
Group-Subgroups Example. Let Γ be a group and Γi be subgroups in Γ
with generating subsets Gi ⊂ Γi , and let Ii (r) be some monotone minorants of
the isoperimetric profiles I◦ (r; Γi , Gi ). Then there is a monotone
P function I(r)
minorizing the isoperimetric profile of Γ defined with |∂|Σ = i |∂|Gi , such that
the Følner function associated to this I is bounded from below by E◦ (Li = Fi (n))
for Fi associated to Ii and with E◦ referring to the partitions of Γ) into the Γi orbits.
Zk -subexample. The above applies to the free Abelian group Zk with the
standard generators g1 , g2 , ..., gk and the corresponding ”coordinate lines” partitions into the orbits of the cyclic subgroups generated by gi ; but the resulting
inequality is only asymptotically sharp.
Similarly, every Cartesian product space X = (X, ×i Γi ) = ×i (Xi , Γi ) satisfies,
(Πincr )
F◦inc,Σ (n; X) ≥
Y
F◦incr (n; X, Γi ).
i
4.5
Lower Bounds on F by E◦ .
Given a partition P of a measure space X and a subset Y ⊂ X, denote by
sup|Y ∩ P | the supremum of the Fubini measures of the slices of P intersected
with Y and by inf |Y ∩ P | the infimum of these measures taken over the slices
that have strictly positive intersection measures with Y .
35
Then, for each number κ > 0, consider subsets Y 0 = Y 0 (κ) ⊂ Y , such that
sup|Y 0 ∩ P | < κ and denote by |P <κ (Y )| the supremum of the measures of these
Y 0.
One can bound |P <κ (Y )| in terms of the distribution of the (Fubini) measures of the P -slices of Y as follows:
If the union Y<λ ⊂ Y of the slices of measures < λ satisfies |Y<λ | ≤ ε|Y | for
some λ ≥ κ and ε ≥ 0, then
[cλκ]
|P <κ (Y )| ≤ ε|Y | + c(1 − ε)|Y | for c = κ/λ.
In fact, the intersection of every Y 0 = Y 0 (κ) with Y<λ has measure ≤ ελ |Y |
for ελ = |Y<λ |/|Y | while the intersection of Y 0 with the complement Y≥λ =
Y \ Y<λ has measure ≤ c(1 − ελ )|Y |. This implies [cλκ] with ελ instead of ε
and, hence, with ε itself, since ε ≤ ελ and c ≤ 1.
Slice Removal Lemma. (Compare Lemma 1 in [9].) Let Pi , i ∈ I, be
partitions of a measure space X, let Y ⊂ X and let κi be positive numbers such
that
X
|Pi<κi (Y )| < |Y |.
i∈I
Then there exists a subset Y ⊂ Y of positive measure, such that
inf |Y ∩ Pi | ≥ κi for all i ∈ I.
Proof. Let Y be a maximal subset in Y with
inf |Y ∩ Pi | ≥ κi for all i ∈ I,
i.e. having all Pi -slices of measure≥ κi for all i. (Such a maximal Y is, in
fact, unique, since the union of two subsets with Pi -slices ≥ κi also has the
slices ≥ κi ). Then every subset in the complement, say Z ⊂ Y \ Y , contains
a point x ∈ Z, such that the slice S = S(x, i) of some Pi passing through x
has |S ∩ Z| < κi . Therefore, Y \ Y◦ decomposes (as in 2.3 B) into the union of
subsets Zi , i ∈ I, where each Zi has sup|Zi ∩ P | < κi , i.e. it has all Pi - slices of
measures < κi . Thus
X
|Y | ≥ |Y | −
|Pi<κi (Y )| > 0
i∈I
and the proof follows.
On the Average Measures of Slices. Denote by Yi = Y (κi ) ⊂ Y the union
of the Pi -slices
P of Y of measures ≥ κi and observe that ∩i Yi ⊃ Y ; thus, the
inequality i Piκi (Y ) < |Y | guarantees that the intersection ∩i Yi is non-empty.
In fact, one only needs for this the total bound on δi = 1 − |Yi |/|Y |, namely
P
i δi < 1, but the latter does not provide any
P lower bound on the cardinalities
of all slices of ∩i Yi . However, if the sum i δi is small, then the Pi -slices of
∩i Yi are ”large on the average” in the following sense.
Let dxi denote the measure on the quotient space Xi = X/Pi associated
to the partition Pi of X and let us regard the Fubini measures of the Pi -slices
36
Rof subsets Y ⊂ X as functions on Xi denoted |S(xi )|(Y ), xi ∈ Xi . Then
|S(xi )|(Y )dxi = |Y | for all Y ⊂ X by the definition of the Fubini measures
Xi
and the average measure of the non-empty (or, rather, of postive measure) Pi slices of Y is (defined as)
Z
|S(xi )|(Y )dxi /|Y /Pi | = |Y |/|Y /Pi |.
Xi
Since the non-empty slices of every Yi have measures≥ κi ,
Z
|Yi |/|Yi /Pi | =
|S(xi )|(Yi )dxi /|Yi /Pi | ≥ κi
Xi
and the average measure of the non-empty Pi -slices of the intersection ∩i Yi is
bounded from below for all i by
| ∩i Yi )|/|(∩i Yi )/Pi | ≥ | ∩i Yi |/|Y /Pi | ≥ κi (1 −
[∩i /Pi )]
X
δi ).
i
Evaluation of F by E◦ . (Compare [9]) The Følner transform for every
family of partitions Pi is bounded from below by the E◦ -function,
(F E◦ )
F (ni ) = F (Fi (ni ); Pi ) ≥ E◦ (ci Fi (εi ni )).
where 0 ≤ ci , εi ≤ 1 are arbitrary numbers satisfying
X
(∗)
(εi + ci (1 − εi )) < 1.
i
Proof. If |∂|Ii (Y ) ≤ |Y |/ni , then the measure of the subset Y<λi ⊂ Y
that is the union of the Pi -slices of Y with measures < λi = Fi (εi ni ) satisfies
|Y<λi | ≤ εi |Y | according to the definition of Fi . Then, according to [cλκ],
|Pi<κi (Y )| ≤ (εi + ci (1 − εi ))|Y |
and the slice removal lemma yields the required bound
F (ni ) ≥ |Y | ≥ |Y | ≥ E◦ (κi = ci Fi (εi ni )).
Product Example. The multivariable Følner function F of the Cartesian
product X of Γ-spaces Xi , i = 1, 2. ..., k, is bounded by the Følner functions Fi
of Xi by
Y
F (ni ) ≥
ci Fi (εi ni ),
i
P
provided i (εi + ci (1 − εi )) ≤ 1, e.g. for εi = 1/2k and ci = 1/(2k − 1). For
instance, if k = 2, then
F◦ (n; X) ≥
n
n
1
F◦ ( ; X1 ) · F◦ ( ; X2 ).
9
4
4
(Sharper inequalities are available for •-convex Fi , see 5.11.)
37
4.6
E◦ -Functions and Group Extensions with and without
Distortion.
Let Γ1 be a normal subgroup in Γ and Γ2 = Γ/Γ1 . Then the above inequality
conjunction with the Schwartz symmetrization shows that
1
4◦
F◦ (n; Γ, G1 ∪ G̃2 ) ≥
1
n
1
n
F◦ ( ; Γ 1 ) · F◦ ( ; Γ 2 , G 2 )
9
4
4
4
where G̃2 ⊂ Γ is some lift of G2 to Γ. Furthermore, if Γ2 is orderable, then
1
4∗
F∗ (n; Γ, G1 ∪ G̃2 ) ≥
1
n
1
n
F∗ ( ; Γ1 ) · F◦ ( ; Γ2 , G2 ).
9
4
4
4
Bound F E◦ with Distorted Subgroups. The above lower bound
F E◦ on the Følner function of a group Γ by those of given subgroups
Γi ⊂ Γ and the E◦ -functions of the family of the partitions of Γ into Γi -orbits
can be sometimes strengthened if (some of) the subgroups Γi are “sufficiently
distorted” in Γ.
Namely, let µi be probability measures supported on Γi ∩ G·mi , for a given
finite generating subset G ⊂ Γ, and Fµi be the Følner functions of Γi associated
to ∂µi . Then F E◦ provides the lower bound on F◦ (n, Γ, µi ) in terms of
Fµi , where, in turn, the G-boundary is bounded from below by m1i · ∂µi for all
i according to the G·m -inequality from 1.4.
If one takes µi uniformly spread over Γi ∩ G·mi , then one can apply the
Displacement Inequality from 1.4. For instance, if the subgroups Γi are infinite
cyclic with the relative growth functions Gi (n) =def |Γi ∩ G·n | n, then one
gains by substituting Li 7→ Gi (Li ) in E◦ .
If one thus capitalizes on distortion of a normal subgroup Γ1 in Γ in 14 ◦ then
n
one replaces F◦ ( n4 ; Γ1 ) in 14 ◦ by 12 G◦◦ ( n2 ) = 12 |Γ1 ∩ (G1 ∪ G̃2 )· 2 | (say for n even)
and obtains the following inequality.
1
8◦
F◦ (n; Γ, G1 ∪ G̃2 ) ≥
n 1
n
1
G◦◦ ( ) · F◦ ( ; Γ2 , G2 ).
18
8 4
4
For instance, polycyclic, e.g. nilpotent non-virtually Abelian groups Γ have
distorted normal subgroups but the above inequality fares no better than (F◦ (n) ≥
1
n
2 G◦ ( 2 ) in these (all?) cases and (reasonably) sharp isoperimetric inequalities
for ”distorted normal extensions“ remain problematic.
Distortion and Isoperimetry of Homogeneous Spaces. Let Γ1 ⊂ Γ be an
arbitrary (not necessarily normal) subgroup. Then the isoperimetric profile of
the homogeneous Γ-space X = Γ/Γ1 is bounded by the profile of Γ. If we want
to take into account the profile and the distortion of Γ1 we may use the action
of Γ × Γ1 on X1 = Γ, where Γ acts from the left and Γ1 from the right. Then
the above inequality remains valid,
F◦ (n; X1 ) ≥
1
n
1
n
F◦ ( ; X) · G◦ ( ; Γ1 ⊂ Γ),
4
4
4
4
where, observe
38
F◦ (n; X1 ) ≤ F◦ (n; Γ × Γ1 ) ≤ F◦ (n; Γ) · F◦ (n; Γ1 ).
Question. What is the (exact if possible) relation between the isoperimetric
profiles of a group Γ and of this Γ regarded as the homogeneous space under
the right/left action of Γ × Γ?
5
Entropy.
Let (X, λ) be a Borel measure space, where λ is regarded as a background measure and where we use the notation |Y | = |Y |λ = λ(Y ) for all Y ⊂ X.
The basic examples are given by countable spaces (X, λ) with the unitary
measures, where all atoms have unit weights (thus, |Y | = card(Y )) and by the
Euclidean spaces with the Lebesgue or with the Gaussian measures λ.
Consider measures µ = f (x)λ for (non-strictly) positive measurable functions f on X and first define the entropy of such a µ where f (x) is constant on
its (essential) support S = supp(f ) ⊂ X by
Z
entλ (µ) = log|S| = log(µ(S)) − µ(S)−1
log(f )dµ,
S
where, observe, f ≡ µ(S)/|S|.
Then, for a general µ = f λ, let |µ|ε = λε (µ) denote the infimum of the
λ-measures of the subsets Sε ⊂ X with µ(Sε ) ≥ (1 − ε)µ(X), where we assume
that µ has finite total mass, |µ| =def |X|µ = µ(X) < ∞.
Take the Cartesian (tensorial) powers (X N , λN = λ⊗N , µN = µ⊗N ) and
with λ⊗N for the background measures on X N . Set
ent(µN − [ε]) = lim inf
N →∞
1
log|µN |ε )
N
and
ent(µ) = entλ (µ) = lim ent(µN − [ε]).
ε→0
Observe that, this entropy is invariant under scaling of µ, that is ent(c · µ) =
ent(µ), while entcλ = entλ + log(c)|µ|.
If µ is a probability measure with a λ-measurable density function f = dµ/dλ
and with the support denoted S ⊂ X, then entλ (µ) ≤ logλ(S) with equality
(only) for µ = λ(S). On the other hand, entλ (µ) ≥ log(supx∈S f (x))−1.
We shall use the above definition only for log-LLN-measures µ, i.e. where
µ = f λ for a λ-measurable function f , such that log(f ) satisfies
The Law of Large Numbers. The µ⊗N measure of the subset
Y (ε, N ) ⊂ X N of the points y ∈ X N , where
Z
1
⊗N
|log(f (y)) −
log(f )dµ| ≥ ε
N
S
satisfies
(LLN )
µ⊗N (Y (ε, N )) → 0 for N → ∞.
39
. For example, LLN holds if |log(f )| is bounded on the support S of µ.
If µ is not log-LLN, one can LLN-regularize it, i.e. approximate by log-LLNmeasures: just cut off the part of S where |log(f )| approaches infinity and then
define the regularized entropy with such an approximation.
Cartesian Additivity of the Entropy. Observe that LLN ensures the additivity of the entropy under the Cartesian product of measure spaces and yields the
celebrated
Boltzman Formula. All log-LLN-measures µ satisfy,
entλ (µ) = log|µ| − |µ|−1
Z
log(f )dµ = log|µ| − |µ|−1
S
Z
f log(f )dλ
S
(for |µ| denoting the total mass µ(X) = µ(S)).
In other words,
the µ-average of log(f ) plus entλ (µ) equals the log of the total mass of µ.
In particular, the entropy of a probability measure µ is expressed by the
Boltzman integral,
Z
Z
1
1
ent(µ) =
log dµ =
f log dλ.
f
f
S
S
This formula is customary taken for the definition of the entropy without
assuming LLN, but only the convergence of the Boltzman integral, possibly to
±∞. This definition is equivalent to the above “regularized entropy” but in all
our applications we can (and do) assume that µ is log-LLN.
5.1
Hölder Inequality via Tensorisation.
We introduce below the Gibbs tensorisation trick and then use it for the proof of
the Shannon inequalities relating the entropy of a measure and its pushforards
under the maps (partitions) in a given family.
Hölder Inequality. The log of the integral
Z Y
fi (x)βi dx
X i∈I
is a convex function of β = {βi } ∈ RI for arbitrary positive functions fi on X.
Proof. The inequality
log
Z Y
X i∈I
!
αi βi
fi (x)
dx ≤
X
Z
αi log
i∈I
fi (x)βi
= log
Y Z
fi (x)βi
αi
i∈I
P
for i αi = 1, αi ≥ 0 is (trivially) true if the functions fi (x) are constant on
the intersection S ⊂ X of their supports (with the equality for functions with a
common support S ⊂ X, where all fi are constant) and the general case reduces
to this by the law of large numbers via the tensorisation.
This argument also shows by how much the inequality
deviates from equality.
Q
Denote µi = gi dx for gi = fiβi and let µ = ( i∈I giαi )dx. Then
40
log
Z Y
giαi dx
≤
X
αi entµi (µ) ≤
X
αi log(µi (X)) = log
i∈I
Y Z
i∈I
αi
gi dx
.
X
The Hölder Inequality can be equivalently stated as follows
let ν be a measure on the linear space X, and let Y be the linear dual to X.
Then the function
Z
Ψ(y) = Ψν (y) = log
exphx, yidν
X
is convex on Y , where the entropy of a measure with the density
function
R
exp(ϕ(x)), x ∈ R equals the derivative ψ 0 (y = 1) for ψ(y) = R exphx, yidx
by the Boltzman formula.
This appears in the Gibbsian thermodynamics as the concavity of the entropy
of the ideal gas and represents a tiny instance of Boltzmann’s and Gibbs’ ideas
(see [20]).
Remarks.(a) The information theoretic rendition of the Gibbs argument is
often presented as a chat between Alice and Bob. (See [11] and references
therein.)
(b) The differential DΨ : Y → X injectively sends Y to X, where
the closure of the image equals the convex hull of the support of µ.
Thus, if X = Y = Rn , then the volume of this hull equals the integral of the
determinant of the Hessian of the (convex!) function Ψ, where the R+ -valued
map
Z
Ψ 7→ M (Ψ) =def
det(Hess(Ψ(y)))dy
Y
1
obeys non-trivial convexity relations: the Minkovski inequality, M n (Ψ1 +Ψ2 ) ≥
1
1
M n (Ψ1 ) + M n (Ψ2 ), and the Alexandrov-Fenchel-Hodge inequality. (See [13] for
a survey and references).
5.2
Entropic Profiles and Stable E◦ Functions of Families
of Partitions.
Given a finite mass measure µ on a Borel measure space X = (X, λ) with a
family of partitions Pi , we denote by µi = µ/Pi = the pushforward of µ to
Xi = X/Pi , call this the Pi -reduction of µ, and write
ent(µ/Pi ) = ent(µi ) = entλi (µi )
for the background measures λi in Xi .
For example, if µ equals the restriction of the background measure λ on X
to a subset Y ⊂ X, then the value of the density function of µi with respect to
the background measure λi on X/Pi at each point xi ∈ X/Pi equals the Fubini
mass of the corresponding Pi -slice of Y .
Denote by µxi xi ∈ Xi = X/Pi the measure f dP (x) on the slice P −1 (xi ) for
the background Fubini measure dP (x) on this slice and f = dµ/dλ and let entxi
be the entropy of µxi with respect to dP (x) on this slice. Define the entropy of
(X, µ) over Xi , also denoted ent(Pi ) as the average
41
ent(Pi ) = µ(X)−1
Z
entxi dµi .
Xi
It is obvious (but significant) that
Entropy is additive.
ent+
ent(P ) + ent(µ/P ) = ent(µ).
Finite Example. Let P be a partition of X, a finite set with the unitary
atoms and take a subset Y ⊂ X. Denote by |P (y)| the cardinality of the P -slice
of Y through y ∈ Y , and observe with the Boltzman (and Shannon in the finite
case) formula that
ent(P |Y ) = log
Y
|P (y)|
|P (y)|
|Y |
.
y∈Y
Entropic Profile. Consider a family P of partions Pi , i ∈ I, of X, where we
usually assume that the single slice partition, corresponding to the map of X to
a single point, is among our P . Every LLN measure µ on X defines the point
e(µ) = {ent(Pi )} ∈ RI ; the set EN T (P) of these points for all µ is called the
entropic profile of P. In what follows we shall evaluate the conical convex hull
of EN T (P) ⊂ RI in the simple cases.
The definition of the entropy and the slice removal lemma from 4.4 imply
the following
Sliced Tensorisation Lemma. Given a finite family P of partitions Pi ,
i ∈ I, of X and an LLN measure µ on X, there exists, for every ε > 0, an
integer N0 = N0 (ε, µ, P) and a subset Y = YN in the Cartesian power X N , for
every N ≥ N0 , such that
Y ⊂ supp(µ⊗N ), where µ⊗N (Y ) ≥ (1 − ε)µ⊗N (X),
⊗N
and the Fubini measures φN
/λ⊗N
of the PiN -slices of Y satisfy,
i =λ
i
N
N
N (ent(PiN ) − ε) ≤ log(φN
i (Pi (y) ∩ Y )) ≤ N (ent(Pi ) + ε)
for all y ∈ Y and all i ∈ I.
Next, observe that the E◦ -functions of Cartesian powers of partitions Pi of
X, satisfy,
N1
N2
N2
N1 +N2
1
E◦ (nN
; PiN1 +N2 )
i ; Pi ) · E◦ (ni ; Pi ) ≥ E◦ (ni
and define
1
N N
E∞ (ni ; Pi ) = lim (E◦ (nN
i ; Pi )) .
N →∞
The above lemma implies the following
Shannon E∞ -Inequality. Let P = {Pi } be a finite family of partitions
on X and µ a measure of finite mass on X = (X, λ). Then the entropies
ent(Pi ) = ent(µ) − ent(µ/Pi ) of Pi with respect to µ satisfy,
ent(µ) ≥ logE∞ (exp(ent(Pi )); P).
42
Remark on Hölder. The tensorisation lemma also implies the Hölder version
of the above inequality.
Let fi ≥ 0 be measurable functions on Xi = X/Pi , let
! p1
Z
fip
|fi |p =
supp(fi )
and let |ΠP fi |1 denote the integral of the product of the pullbacks of fi to X.
Then
|ΠP fi |1 ≥ E∞ (|ΠP fi |1 /|fi |pi ; P)
for all {pi } ∈ RI+ .
If all Pi are single slice partitions, this reduces to the Hölder inequality from
5.1 with positive pi (and with no entropic correction term).
5.3
Shannon Inequalities for the Coordinate Line and
Plane Partitions.
Let (X, λ) = ×i (Xi , λi ), i = 1, 2, ..., k. Then the partitions Pi of X into the
”coordinate lines“ with the slices isomorphic to Xi and corresponding to the
projections Pi : X → Xî = (Xî , λî ) = ×j∈I\{i} (Xj , λj ) satisfy
Sh1
ent(µ) ≥
X
ent(Pi ),
i
or, equivalently,
ent(µ) ≤
1 X
ent(µ/Pi )
k−1 i
for all measures µ on X. Furthermore, the partitions PJ of X into the fibers of
the projections X = XI → XI\J = ×i∈I\J Xi (with ”J-plane“ slices representing
XJ = ×i∈J Xi ) satisfy
Shα
ent(µ) ≥
X
αJ · ent(PJ )
J⊂I
for all partitions of unity αJ of I (see 4.3).
Proof. Here, obviously, E∞ = E◦ and the above applies.
Loomis-Whitney Inequality. This is an upper bound on |Y | = λ(Y ) for
subsets Y ⊂ X in terms of the background measures of Y /Pi , (assuming these
are measurable) written as if it were a lower bound,
Y
|Y | ≥
|Y |(λî (Y /Pi ))−1 .
i
This follows from the Shannon Inequality, since entλî (µi )) ≤ λî (supp(µi ))
and ent(µ) = log(λ(Y )) for µ = λ|Y .
Similarly one derives the Shearer Inequality that is the bound on log|Y | by
log|Y | − log(λJˆ(Y /PJ ) substituting ent(PJ ) in Shα . (The role of the entropy in
such inequalities was pointed out to me by Noga Alon.)
43
If Xi are countable sets with the atoms of unit weights, then the Shannon inequality for subsets Y ⊂ X = ×i Xi with the restricted product unitary
measures reads,
Combinatorial Shannon Inequality for the Coordinate Line Partitions. Let |Pi (y)|, y ∈ Y , denote the cardinality of the Pi -slice of Y through y.
Then the geometric means
 |Y1 |

|M Pi | = 
Y
|Pi (y)|
y∈Y
satisfy
Y
|M Pi | ≤ |Y |.
i
Harper Inequality. The Shannon inequality, when applied to the vertex
set X of the edge graph of a Euclidean n-cube with the edges for slices, says
that
the vertex and the edge numbers of every subgraph Y in the cubical graph
satisfy,
(4N )
Nvert ≥ 4Nedg /Nvert .
For example, if all vertices in Y have the valency (degree) at least d, then |Y | ≥
2d .
Another corollary of the combinatorial Shannon inequality is the following
(well known) relation between the three numbers: the cardinality |Y |, the number N of the slices of Y with respect to all Pi and the sum C of the cardinalities
of all these slices.
AB -Inequality. Let A = C/N and B = C/|Y |. Then
|Y | ≥ AB .
Proof. Since the function ss is log-convex, log(ss )00 = 1/s,
AB ≤
Y
|S|
|S| |Y | ≤ |Y |,
S
where the product is taken over all slices S of the partitions Pi .
5.4
Strict Concavity of the Entropy and Refined Shannon
Inequalities.
A probability measure µ on X1 × X2 can be regarded as a family of probability
measures µx1 on X2 parametrized by x1 ∈ X1 , where the density fx1 (x2 ) of
(almost) every measure µx1 on X2 equals
R the restriction of the density of µ to
x1 × X2 ⊂ X1 × X2 divided by p(x1 ) = X2 fx1 (x2 )dλ2 .
The Shannon inequality written as ent(µ/P2 ) ≥ ent(P1 )(= ent(µ)−ent( µ/P1 ))
says that the entropy is a concave function on the space of probability measures
on X2 , since the measure µ2 on X2 , that is the pushforward of µ, equals the
44
p(x1 )-weighted convex combination of the probability measures µx1 , while the
entropy is (defined as) the corresponding convex combination of the entropies
of µx1 .
In fact, the entropy is strictly concave as follows from the Boltzmann formula
and the strict convexity of the function t · log(t). (This is the common way
for deriving the Shannon inequality). Then the quantity ent(µ) − ent(P1 ) −
ent(P2 ) ≥ 0 tells us how far µ is from equilibrium, i.e. a probability measure µ0
on X1 × X2 , for which the probability measures µ0x1 on X2 are mutually equal
for all x1 ∈ X1 , or equivalently all µ0x2 on X1 are equal.
Here is another characteristic of (non-)equilibrium for measures µ on product
spaces X = ×i Xi , i ∈ I.
The index set I tI, (disjoint union of I with itself) and, hence, the Cartesian
power X 2 of X, is naturally acted by the Mendelian recombination group ZI2 =
(Z/2Z)I generated by |I| coordinate involutions on I t I and/or on Xi × Xi for
all i ∈ I. By strict convexity, a measure µ on X is at equilibrium, where (by
definition if you wish) all Shannon inequalities Shα becomes equalities, if and
only if the measure µ⊗2 on X 2 is invariant under ZI2 and (where, observe, µ⊗2
is invariant under the diagonal involution on X 2 for all µ on X.)
We introduce the entropic displacement of µ⊗2 by z,
1
1
|µ⊗2 − z(µ⊗2 )|ent =def ent( (µ⊗2 + z(µ⊗2 )) − (ent(µ⊗2 ) + ent(z(µ⊗2 ))) ≥ 0
2
2
and then identify involutions z ∈ ZI2 with subsets J ⊂ I by
z ↔ J = J(z) = supp(z) ⊂ I
where the support of z is defined by z(i) 6= i.
The composition of involutions corresponds to the symmetric difference of
subsets that we denote J1 · J2 =def (J1 ∪ J2 ) \ (J1 ∩ J2 ). We also abbreviate by
writing
|J|ent (µ) = |µ⊗2 − z(J)(µ⊗2 )|ent ,
where, observe, |J|ent (µ) = |J ⊥ |ent (µ), for J ⊥ = I \ J.
A measure µ on X satisfies the equality
ent(PJ ) + ent(PJ ⊥ ) = ent(µ)
if and only if µ⊗2 is z(J)- (or,equivalently z(J ⊥ ))-invariant; this is also equivalent to
|J|ent (µ) = 0.
Since the entropy is strictly concave, the function |J|ent (µ) of J ⊂ I satisfies
some triangle-type inequalities,
(∆)
|J1 · J2 |ent (µ) ≤ ∆µ (|J1 |ent (µ), |J2 |ent (µ)),
where ∆µ (0, 0) = 0 for all µ and ∆µ (a, b) is uniformly continuous in (a, b) with
the modulus of continuity δ depending on µ. Moreover, δ is uniformly bounded
on certain (compact in a suitable sense) classes of measures µ.
45
For example, if the density function f of µ satisfies
Z
|log(f (x))|dµ ≤ const < ∞,
X
then δ is bounded by some universal δconst as a simple continuity argument
shows.
This is useful, for instance, if log(f (x)) ≤ 0, e.g. if X is a discrete space
with unitary atoms, where (∆) becomes a relation between the entropies of PJ
depending only on ent(µ),
(∆)
|J1 · J2 |ent (µ) ≤ ∆ent(µ) (|J1 |ent (µ), |J2 |ent (µ)),
for some function ∆e (a, b) that is continuous in a, b and e and such that ∆e (0, 0) =
0.
Remarks. (a) All this is, apparently, well known but I could not find a
reference; nor do I know a specific sufficiently “elegant” ∆e (a, b). I guess, there
are sharp “mixed symmetric mean inequalities” for measures on ×Xi similar to
the classical Muirhead’s inequalities, such as the mixed discriminant inequality
of Alexandrov (that is GL(k)- rather than just Sk -symmetric).
(b) The above generalizes to the Cartesian powers X N with the Cartesian
products of I-copies of the permutation group SN acting on it. The resulting
inequalities become, in a sense, asymptotically sharp for N → ∞ due to the law
of large numbers (applied to convolution of measures on the spaces of measures).
A possible framework for this is suggested by the Mendelian dynamics on
the space of measures on the disjoint union X t = tN X N of the the Cartesian
powers of X = ×i∈I Xi , where, observe, X N = ×i XiN , and where this equality
is defined (only) up to the action of the Cartesian power of the permutation
group, (SN )I acting on X N . Every partition of unity αJ where all α are rational
with the common denominator, say αJ = nJ /d, defines a bijection (equality)
×J XJnJ = X d for XJ = ×i∈J Xi and similarly the equalities ×J XJnNJ = X dN for
all N .
The Mendel map F sends measures µN on X N to measures F (µN ) on the
tensor products of the J N -reductions of µN to XJ N for all subsets J ⊂ I. This
F is accompanied by the power map µN 7→ µ⊗d that should be thought of
as ”multiplication by the scalar = d” in the some kind of log of the tensor
1
algebra. Then the ”ideal dth root” F ⊗ d of the Mendel map apparently (I
did not check this) exponentially contracts (in a suitable metric) the space of
”graded measures” {µN } to the subspace of the equilibrium (i.e. tensor product)
measures (as in the
Mendel-Robbins-Geiringer theorem, (see [17] and references therein) and
satisfies d1 ent(F (µN )) ≥ ent(µ) in the agreement with the Shannon-Shearer
inequality.
Question. What is a comprehensive formalism that would fully reflect the
(SN )I -symmetry along with the N -grading and that would also embrace the
Brascamp-Lieb and the Lieb-Ruskai inequalities below?
(c) The Hessian of the entropy on the space (affine simplex) M of probability
measures, being positive definite by the Shannon inequality, defines a (non- complete) Riemannian metric on M , called Fisher-Rao-Kramer (Antonelli- Strobeck,
Svirezhev-Shahshahani, Karlquist) metric, (also underlying the Einstein-Onsager
46
relations as was explained to me by Alexander Gorban) that, by the Boltzmann
formula, equals the spherical metric induced by the log-radial (compare 2.2) pro√
jection (pi ) 7→ ( pi ) of the unit n-simplex ∆n ⊂ Rn+1 of probability measures
to the unit sphere S n ⊂ Rn+1 .
√
The meaning of (pi ) 7→ ( pi ) becomes clearer with the the moment map
Cn+1 → Cn+1 /Tn+1 = Rn+1
⊃ ∆n
+
for the coordinate-wise (Hamiltonian!) action of the torus Tn+1 on Cn+1 , where
the Riemannian quotient metric on Rn+1
= Cn+1 /Tn+1 identifies with the Hes+
P
n+1
sian of
pi log(pi ) on R+ ⊃ ∆n .
Questions. Is there a geometric picture where this unitary symmetry of the
Fisher metric is seen simultaneously with the tensorisation property (definition)
of the entropy?
What are other ”entropy type” functions associated to Kähler manifolds
with isometric actions of compact groups?
5.5
On Sliced Spaces.
A sliced set is a set X with a distinguished family S ⊂ 2X of subsets S ⊂ X,
called slices. More generally “a slicing” of a measure space X is given by a
measurable map X̃ → X, where X̃ is a measure space with a distinguished
partition P̃ , where the map X̃ → X is one-to-one on (almost) every slice of P̃ .
(Alternatively, a sliced structure is a measure on 2X with some regularity to it.)
Here is a generalization of the combinatorial Shannon inequality for pairs of
partitions to sliced spaces.
No-∆ Inequality. Let X be sliced by subsets S ∈ S ⊂ 2X , such that,
(a) every point of X is contained in at most two slices;
(b) every two non-equal slices intersect at a single point if at all.
If S contains no triple of pairwise intersecting slices, then every finite subset
Y in X satisfies the combinatorial Shannon inequality,
Y
|Y ||Y | ≥
|Y ∩ S||Y ∩S| .
S∈S
Proof. Consider the 2-chains (y0 , S1 , y1 , S2 , y2 ) in Y , where the pairs of
points (yi−1 , yi ) are contained in the slices Si for i = 1, 2 and where S1 6= S2 .
The geometric mean G0 of the number of such chains over y0 ∈ Y , i.e, the
geometric mean of the number N0 (y) of the chains starting at y ∈ Y is ≥ than
the geometric mean G1 of the number N1 (y1 ) = |S1 | · |S2 | of the chains with the
middle point y1 ∈ Y by the geometric-arithmetic mean inequality.
Since the number of the chains between every pair of points (y0 , y2 ) in X is
≤ 2 by our assumptions on S, we get our inequality,
! |Y1 |
! |Y1 |
|Y | ≥
Y
y
N0 (y)
= G0 ≥ G1 =
Y
y
N1 (y)
! |Y1 |
=
Y
|Y ∩S|
|Y ∩ S|
.
S∈S
Questions. What of this kind would generalize Shannon for more than two
partitions? What are further “chain uniqueness” and/or “no short cycle” conditions leading to further inequalities? In particular, if a certain co-entropic
47
inequality holds for all subsets Y of cardinality ≤ N , for large N depending on
the type of the inequality, does it then hold for all finite subsets Y ? Is Gibbs’
tensorisation helpful for such questions?
Notice in this regard that the entropic profile of a sliced space is monotone
decreasing under slice-wise injective maps ψ : X1 → X2 . In fact, a simple
computation shows that the decrease of the entropy under ψ is greater than the
average entropic decreases of the slices of the two spaces for all measures µ on
X1 ,
X
X
ent(µ) − ent(ψ∗ (µ)) ≥
ent(µ|S1 ) −
ent(ψ∗ (µ|S2 ).
S1 ∈S1
S2 ∈S2
Remark. There are many homogeneous sliced spaces X, where one would
like to know the relation between the cardinalities of subsets Y ⊂ X and their
slices but a comprehensive answer may be out of reach.
For example, some relations of this kind in affine and projective spaces X
sliced by their subspaces appear in combinatorial number theory, e.g. in the van
der Waerden and the Szemerédy theorems on arithmetic progressions, where one
uses something more complicated than the Shannon inequality.
Also many (especially homogeneous and symmetric) Riemannian manifolds
are naturally sliced by some families of submanifolds, e.g. by geodesics (when
these are properly embedded) and by higher dimensional (geodesic) submanifolds. The problem of determining the co-entropic (as well as isoperimetric)
profiles for these “slicings“ remains open in most cases.
P N -Tensorisation. One can slightly improve the E∞ -ent-relation by taking
into account the following.
1. The Cartesian power P N of the family of partitions Pi , i ∈ I, consists
of |I|N partitions indexed by maps {1, 2, ..., N } → I (our definition of E∞
takes into account only the |I| partitions PiN corresponding to the constant
maps), where the set I N is naturally acted upon by the permutation group SN
and where Q
the orbits of the corresponding
partitions of X N are regarded as
P
Ni
monomials Pi over all {Ni } with
Ni = N .
2. The fully tensorised E◦ -function is submultiplicative while the entropies
of partitions are multiplicative.
With this in mind, we define E⊗ (Li ) ≥ E∞ (Li ) as we did for E∞ , except
that now we take
subsets Y ⊂ X M ·N , where the slices of the partitons
Q the
Mi N
represented by ( Pi ) have the Fubini measures bounded from below by
Q
Q Mi
i N
( LM
for all monomials
Pi , and take the maximal function E⊗ that
i )
minorizes the measures of all such Y for M, N → ∞ with N M .
(The smaller the class of subsets Y we use, the greater is the resulting E◦ function, where one may further limit the class of Y by requiring the symmetry
of Y ⊂ X N under the SN -action. On the other hand, if one limits the class of
competing E, then the outcome becomes smaller. Yet, one loses little in the
present context if one maximizes over all convex functions E.)
Then we see as earlier that
ent(µ) ≥ logE⊗ (exp(ent(Pi )); P).
48
5.6
Fiber Products, Fubini Symbols and Shannon Expansion Inequalities.
Call (typically, surjective) measurable maps X → Y partitions of X over Y
with the slices in X thought of as the fibers. Given two spaces X1 and X2
partitioned over Y , denote by X1 ×/Y X2 ⊂ X1 their fiber product, that is the
set of the pairs (x1 , x2 ) ∈ X1 × X2 that lie over the same point in Y .
The set X1 ×/Y X2 ⊂ X1 carries a unique Fubini measure, such that its
natural projections to X1 and X2 are measurable, and consistent with the Fubini
measures in the slices.
The space X1 ×/Y X2 ⊂ X1 , comes along with two fiber maps:
the first one is induced from P1 , i.e. is given by the map (P1 , Id2 ) : X1 ×/Y
X2 → X1 × X2 , where all slices isomorphically go to to slices of P1 , and the
second one is similarly induced from P2 .
Given a family of partitions of a space X, or, better, a small category C
of measure spaces Xi (where we do not distinguish any X at this point) and
measurable maps Pij : Xi → Xj , we compose the above operations and thus
generate a small category F = F(C) of measurable spaces and maps that is
built inductively, starting from C by taking fiber products for new objects and
the fiber maps for the morphisms.
We always include into this category the identity map of each space, corresponding to the partition into points as well the constant map, where the only
slice is the space itself.
Examples. (1) If there is a single partition P1 : X → X1 of X then all
one essentially can do is to take the iterated graph of P1 denoted X k /P1 =
X k /X1 that is the set of points in X k that go to the same slice in X under
the k projections X k → X. This X k /X1 comes along with k (isomorphic but
different) partitions, each of them is indiced from P1 .
(2) Given a sequence of partitions Pi , i = 1, 2, ..., k, of X, define the space
Z(k) of k-chains of points x0 , x2 , x2 , ..., xk , where every two consecutive points
xi−1 , xi lie in some slice of Pi . This space is constructed by taking the consecutive fiber products of the graphs X ×Pi X over X and thus it carries the Fubini
measure on it. This Z(k) is mapped to X × X by sending the two end points
of chains of points back to X but this map is not necessarily measurable.
For example, if X = Γ is a locally compact group and Pi are partitions into
the orbits of closed subgroups Γi ⊂ Γ, then this map is measurable if the group
product map ×i Γi → Γ is open and countable-to-one.
The fiber product is neither a commutative nor an associative operation:
a subset Y ⊂ Π = ×i Xiki can be decomposed into a fiber product in many
different ways (if at all). However,
the measures on Y coming from these decompositions are all equal by the
Fubini theorem and can be described with the following
Fubini Symbols. Let A = A(F) = A(C) be the (Grothendieck) Abelian
group generated by the symbols [X] for all objects X in F with the relations
(F)
[X1 ×X3 X2 ] = [X1 ] + [X2 ] − [X3 ].
for all fiber products in F. Clearly, A is generated by (the symbols of) the
objects in F.
49
Example.
The space Z of the k-chains of partitions of X over Xi has [Z] =
P
[X] + i ([X] − [Xi ]). In particular, the k-iterated graph Z of X over Y has
[Z] = [X] + k([X] − [Y ]).
Expansion Coefficients of Coordinate Maps. Consider two objects Z1 to Z2
in F realized by subsets in the Cartesian products of Xi , say in ×i Xiki and ×Xili
where li ≤ ki and assume that some coordinate projection p : ×i Xiki → ×Xili
sends Z1 to Z2 .
Example. The above space of k-chains c = {x0 , x2 , x2 , ..., xk } in X goes thus
to X × X by c 7→ (x0 , xk ).
In general, such coordinate maps p between Z’s are not measurable and
the pushforward λ∗ of the background measure λZ1 from Z1 to Z2 may be
everywhere infinite even for measurable p.
We are concerned with the case when λ∗ is locally finite, that is λ∗ =
δ(z2 )λZ2 for a measurable function δ on Z2 and we say in this case that p
is ∆-expanding for a positive constant ∆ if δ(z2 ) ≤ ∆−1 (almost) everywhere
on Z2 . Whenever such a map exists we write, the corresponding
log-inequality,
[Z2 ] ≥ [Z1 ] + log(∆).
Examples. (a) A map between countable spaces with unitary atoms is expanding, i.e. 1-expanding, if and only if it is injective.
(b) An injective smooth equidimensional map between Euclidean spaces is
∆-expanding if its Jacobian ≥ ∆.
Shannon Expansion Theorem. Let C be a small category of Borel measure spaces with the maximal object X, where all other objects Xi , i ∈ I, are
quotients X/Pi for partitions Pi of X. Let p; Z1 → Z2 be a ∆p -expanding map
in the fiber-product category F generated by C and write the corresponding loginequality in the symbols [X] and [Xi ] (that generate the group A),
X
k[X] +
ki [Xi ] ≥ lp
i
for the (positive and/or negative) integers k and ki defined by (the Fubini symbols of ) (Z1 , Z2 )) and lp = log∆p . Then every measure µ on X satisfies
(ExSh)p
k · ent(µ) +
X
ki · ent(µi ) ≥ lp
i
for the pushforward (reduction) measures µi = µ/Pi on Xi .
Proof. The claim is obvious if µ equals the restriction of λ to a Borel subset
Y ⊂ X with λ(Y ) < ∞ and such that the Pi -slices of Y have constant Fubini
measure for every partition Pi , since the Fubini measures are multiplicative
for fiber products. Then the general case follows by the Gibbs tensorisation
argument as earlier. QED
5.7
Weighted Loomis-Whitney-Shearer and BrascampLieb Inequalities.
Each (ExSh)p is a linear inequality in the variables e and ei representing the
entropies, where the Shannon expansion theorem says that
50
the closed convex hull Hent = Hent (X, Pi ) of the set of the vectors (ent(µ), ent(µi ))
in the Euclidean space E+ = R|I|+1 for µ running over all measures µ is contained in the intersection Hex of the closed half spaces defined by all (ExSh)p .
Moreover, since the expansion condition passes to the Cartesian powers (X N , PiN ),
st
the (stabilized and possibly enlarged) closed convex hull Hent
⊂ R|I|+1 of the vec1
N
N
tors N (ent(µ), ent(µ/Pi )) for all measures µ on X , is also contained in Hex .
st
Observe that the convex subsets Hent
⊂ Hex ⊂ R|I|+1 are unbounded and
st
only in rare cases Hent ⊂ Hex . On the other hand, the two sets are often
st
parallel at infinity: if a linear form is bounded from above on Hent
then it is
also bounded on Hex .
Here is the simplest (ExSh).
1. Shannon and Loomis-Whitney Inequalities for Expanding Families of Partitions. If the (two ends) map from the space Z(k) of chains to
X × X for partitions P1 , ...Pi , ..., Pk (see above Example 2) is expanding, then
Pi satisfy the Shannon inequality
X
ent(Pi ) ≤ ent(µ)
i
for all measures µ on X and, therefore, the Pi -projections of every Y ⊂ X
satisfy the Loomis-Whitney inequality,
Y
i
λ(Y )
≥ λ(Y ).
λi (Pi (Y ))
st
This inequality is sharp, i.e. Hent
= Hex , provided the spaces X/Pi are not
monoatomic.
Example. The ”expanding“ assumption is (obviously) satisfied for the orbits
(cosets) partitions of a simply connected nilpotent Lie group X = Γ by the
orbits (cosets) of connected subgroups Γi if the Lie algebras of Γi are linearly
independent and span the Lie algebra of Γ, provided the product map ×Γi → Γ
has the Jacobian = 1 at the origin and the same is true for unipotent algebraic
groups over the locally compact groups.
But the issuing Shannon and Loomis-Whitney inequalities are non-sharp for
non-commuting subgroups Γi (i.e. ≤ can be replaced by <), where one expects
significantly stronger inequalities (due to non-commutativity) for the measures
that are ”sufficiently uniformly spread“ over Γ.
2. Shannon and Shearer Inequalities for Weighted Coordinate Partitions. Let PJ be the coordinate plane partitions of X = XI = ×i∈I Xi into
the slices that are the fibers of the projections X → XI\J = ×i∈I\J Xi , J ⊂ I,
where the background measures λJ on XJ = ×i∈J Xi are not assumed to be
product measures but rather satisfy the following assumption.
The measures λJ on XJ = ×i∈J Xi are greater than the λI -Fubini measures
on the PJ -slices: the (bijective!) coordinate projection of every PJ -slice S ⊂ X
to XJ is expanding with respect to the λI -Fubini measure on S and λJ on XJ .
This condition implies that all bijective coordinate maps between the fiber
products of XJ are expanding and the proof follows by the inclusion-exclusion
principle. Thus
the coordinate inequalities from 5.3 remain valid under the above expansion
assumption.
51
In particular, one has
Shearer-Loomis-Whitney Projection Inequality. The measures of the
projections PJ (Y ) ⊂ XJ satisfy
Y
(P◦ )
(λJ (Y ))αJ ≥ λI (Y )
J⊂I
for all Y ⊂ X and all partitions of unity αJ .
3. Euclidean Brascamp-Lieb Inequalities. Let Pi , i = 1, 2, ..., n, be
partitions of Rk into parallel non-coordinate affine subspaces. Then
the entropies of Pi with respect to an arbitrary probability measure µ on Rk
satisfy the inequality
!
X
ki · ent(Pi ) − ent(µ) ≥ l
i
for some real constants ki ≥ 0 and −∞ < l < ∞ (depending on Pi but not on µ,
where one may always normalize to l = 0 by rescaling the background measure
λ be exp(−l)) if and only if they satisfy this inequality (with these very ki and
l) for all Gaussian measures ν on Rk . This is usually stated in the (equivalent)
Hölder form (see [12] and references therein).
The Shannon Expansion theorem is weaker in the Euclidean case: it only
delivers qualitative form of these inequalities, saying, in effect, that
st
the intersections of Hent
and Hex , with the (convex) subset e = −1, ei ≥ 0 ⊂
k
R are parallel at infinity.
Sketch of the proof. All fiber products are Euclidean spaces and the coordinate projections are all linear; those which are bijective are ∆-expanding
for various ∆ and these deliver a set of (ExSh)-inequalities that are, by simple
linear algebra, equivalent, up to l’s, to the Brascamp-Lieb inequalities.
This also applies to families of partitions into orbits of locally compact
Abelian groups and nilpotent (and locally compact unipotent) Lie groups by
families of subgroups, but , as in the above Loomis-Whitney case, this is less
satisfactory (apart from being non-sharp even in the Abelian case) since the
effects of non-commutativity (potentially, strengthening such inequalities) are
neglected.
One can slightly improve such inequalities similarly by adding to our category maps Π permuting the coordinates of product spaces, where the additional
terms in the Shannon inequalities come from non-invariance of measures on (and
/or subsets in) such products.
These Π may generate infinite groups of transformations of such products
that admit no finite invariants measures.
Example. Let Pi , i = 1, 2, 3 be the partitions of R2 into the three families
of parallel lines in general position. Then the group generated by the three
involutions of R4 corresponding to the three splitting of R2 into the product of
lines, contains a free subgroup. (But the resulting improvement of the Shannon
inequality, even in this simple case, remains unsharp unlike the corresponding
Brascamp-Lieb inequality.)
Riemannian Remark. The ∆-expansion condition makes sense for the exponential maps for Riemannian manifolds X, where, for example, it is satisfied
52
with ∆ = 1 by the complete simply connected manifolds with the sectional
curvatures K ≤ 0.
The geometric counterpart to the Loomis-Whitney inequality is a bound on
the volume |Y | of a Y ⊂ X by some function of the (Liouville) measure |S| of
the set S of geodesics that meet Y in X. If Y is convex, i.e. it meets every
geodesics over a (connected and simply connected ) segment, then |S| = |∂(Y )|
with a suitably normalized Liouville measure and |S| ≤ |∂(Y )| in general).
Non-sharp bounds follow by the Fubini theorem (as in the Coulhon-SaloffCoste case) but a sharp inequality is available only for X = Rn by Almgren’s
variational argument [1] that applies, to Riemannian manyfolds X, (say, complete with bounded geometry, for safety) such that
the integral mean curvature M of the ”exposed” locally convex part E of the
0
boundary of every Y admits a lower bound M ≥ IX
(|∂(Y )|), where a point
y ∈ ∂(Y ) is called exposed if the set Cy of the geodesic rays s that start at y and
meet Y at some y 0 6= y, contains no pair (s, −s), and where the lower bound
on M depends on the function I 0 – a kind of a derivative of the isoperimetric
profile.
If X = Rn , then such bound on M is obtained by taking the convex hull H
of Y , where the total Gauss curvature of ∂(H), (that is independent of H ⊂ Rn )
minorizes the (obviously “homogenized“) mean curvature M , since the Gauss
curvature of ∂(H) vanish outside E = ∂(Y ) ∩ ∂(H). Thus,
among all subsets Y ⊂ Rk of given k-volume, the round balls minimize the
(Liouville) measure of the set of straight lines that intersect Y . (If Y are convex,
this follows from the ordinary isoperimetric inequality).
5.8
Linearized Shearer-Loomis-Whitney Inequalities.
Let Xi , i ∈ I, be vector spaces over some field and denote by XJ , J ⊂ I,
the tensor product of Xi over J, i.e. XJ = ⊗i∈J Xi . Define the J-reduction
YJ ⊂ XJ of a linear subspace YI ⊂ XI as the minimal subspace in XJ , such
that YJ ⊗ XI\J contains YI .
To see it better, take the tensor product ZI = XI ⊗ Y 0 for some linear space
0
0
Y take a vector z = ZI and let zJ : XI\J
⊗ Y 0 → XJ be the homomorphism
0
corresponding to z under the canonical isomorphism XI ⊗ Y 0 = Hom(XI\J
⊗
0
Y 0 , XJ ), where XI\J
denotes the linear dual of XI\J .
Then the J-reduction of the image Y ⊂ XI of zI equals the image of zJ in
XJ ; thus, rank(YJ ) = rank(zJ ).
Tensorial Reduction Inequality. The ranks of the J-reductions of every
linear subspace Y ⊂ XI = ⊗i∈I Xi satisfy
(P⊗ )
Y
(rank(YJ ))αi ≥ rank(Y )
J⊂I
for an arbitrary partition of unity {αJ } on I.
Proof. Take bases Bi in each Xi , let BI = ⊗i∈I Bi be the corresponding
basis in XI and project Y to a coordinate plane P along the complementary
coordinate plane. Clearly, |J|T ≤ |J|Y for all Y ⊂ XI and all J ⊂ I. Since
there is a P , such that the coordinate projection Y → P is bijective, the above
(P◦ ) applies and yields (P⊗ ).
53
Remark. Given a k + 1-linear form y = y(x0 , ..., xk ), each splitting J of the
variables xi into two groups defines a bilinear form yJ (on the tensor product
corresponding to the splitting) and (P⊗ ) becomes an inequality between the
ranks of the forms yJ .
A particular instance of this (where one may have stronger inequalities) is
the cohomology (sub)algebra X of an (algebraic) manifold V , where y(xi ) equals
the value of the cup-product of xi on the fundamental class of V . (See [16] for
topological applications of such inequalities.)
5.9
Orbit and Lieb-Ruskai Inequalities for Representations of Compact Groups.
Consider a finite dimensional linear representation Z of a group Γ with a given
family of subgroups ΓJ , J ∈ J , take a vector z ∈ Z and let |ΓJ (z)| denote the
rank of linear span of the ΓJ -orbit of z.
If Γ is the Cartesian product, Γ = ×i Γi , i ∈ I, and ΓJ = ×i∈J Γi we have
the following
Linearized Shearer Inequality. If the representations of Γ and all ΓJ are
semisimple (over the algebraic closure of the ground field), then, for an arbitrary
partition of unity {αJ } on I, the rank of the linear span of the Γ-orbit of every
point z ∈ Z satisfies,
(P∗ )
|Γ(z)| ≤
Y
|ΓJ (z)|αJ .
J⊂I
Proof. If Z is a multiple of an irreducible representation, i.e. Z = (⊗i Xi )⊗Y 0
for irreducible representations Xi of Γi , then the linear span of the ΓJ orbit of
z equals the tensor product of ⊗i∈J Xi with the image of the corresponding
homomorphism zJ0 : (⊗i∈J Xi )0 → (⊗i∈I\J Xi ) ⊗ Y 0 and the above tensorial
inequality applies.
Then the general case follows by applying (P◦ ) from 5.7 to the Cartesian
product of the sets Ri (earlier denoted Xi ) of irreducible representations of Γi
with the measures assigning to each representation its rank weighted on RJ by
the multiplicity with which the irreducible representation rJ of ΓJ enters the
span of the orbit ΓJ (z).
Remarks.(a) The above applies to the representations of finite groups over
the fields of the characteristic prime to |Γ| and also to the real and complex representations of compact groups; it is unclear if the semisimplicity assumptions
can be removed or relaxed.
(b) The inequality (P∗ ) over C admits an entropic (and, probably, a Hölder)
refinement with the
Lieb-Ruskai Inequality for Quantum Entropy. The quantum (von Neumann)
entropy is defined for positive self adjoint operators with unit traces on, say,
finite dimensional, Hilbert spaces Z, or equivalently for positive (semi definite)
quadratic forms µ on Z, similarly to the Boltzmann entropy .
A quadratic form µ is regarded as a function on the linear subspaces in Z
that assigns to each Y ⊂ Z the trace of µ restricted to Y , where, observe, this
function is additive under the spans of mutually orthogonal subspaces. If PY is
54
the orthogonal projection onto a subspace Y ⊂ Z and µ = dim(Y )−1 PY , then,
by definition, ent(µ) = log(dim(Y )).
Then, in general, one passes to the tensorial powers (Z ⊗N , µ⊗N ), takes the
ε-approximation (in the trace norm) of µ⊗N by a ”measure” µN
ε associated to
a projection on a subspace YN ⊂ Z ⊗N of maximal (depending on ε dimension)
and set
ent(µ) =def
1
ent(µN
ε ).
N →∞ε→0 N
lim
The law of large numbers (for the spectral measure of µ) shows that ent(µ)
equals the entropy of the Boltzman entropy of the spectral measure, where the
background measure (corresponding to the background Hilbert structure on Z)
is that of a discrete space with all atoms of unit weight.
Given such a ”measure” on a tensor product of Hilbert spaces, L = L1 ⊗ L2 ,
one defines its reductions µi on Li , i = 1, 2, say µ1 , by taking the traces of
the corresponding self-adjoint operators over L2 . Equivalently, one can average
µ over a compact group Γ2 unitarily and irreducibly acting on L2 , and thus,
on L, where the averaged ”measure”, say µ01 on L, can be uniquely written as
µ01 = µ1 ⊗ λ2 for λ2 corresponding to the scalar operator on L2 with trace = 1
that is |L2 |−1 · id.
Lieb-Ruskai Strong Superadditivity.. Let µ correspond to a positive
self-adjoint operator with trace(µ) = 1 (”probability measure”) on the tensor
product of three finite dimensional Hilbert spaces, L = L1 ⊗ L2 ⊗ L3 . Then the
entropies of its reductions to L1 ⊗ L2 , L2 ⊗ L3 and L2 satisfy,
ent(µ12 ) + ent(µ23 ) ≥ ent(µ2 ) + ent(µ = µ123 ).
(See [26], [5] and references therein).
This inequality, (trivially, by the inclusion-exclusion principle) implies the (
weighted with partitions of unity ) Shannon type inequalituies for tensor products of (more than three) Hilbert spaces and thus for Hilbert spaces acted upon
by Cartesian products of compact groups. This sharpens the above reduction
inequalities when the underlying field F = C. Furthermore, this generalizes to
the actions of infinite groups Γi , on l2 (Γ = ×i Γi ), where the entropy is defined
via the von Neumann Γi -dimensions ( [23]). But this does not seem (at least,
at the first glanc) to yield Loomis-Whitney projection (reduction) inequalities
(nor isoperimetric inequalities) for finite dimensional subspaces D ⊂ l2 (Γ) for
non-Abelian Γi .
Questions. (a) The quantum entropy is a concave function on the space
Mn of positive selfadjiont operators on Rn (by the superadditivity of the quantum entropy proved by Landford prior to the Lieb-Ruskai theorem) and the
geometry of the corresponding ”quantum ” (Fisher-)Kubo-Mori-Bogolubov metric and similar Bures-Uhlmann and Hasegawa-Nagaoka-Pets metrics (see [24])
along with the symmetric Riemannian metric on Mn = GLn /On , probably, tell
you something else about (relations between the quantum entropies of) various
reductions. Furthermore, the relative entropy defined on the pairs of positive
selfadjoint operators by T race(µlog(µ) − µlog(ν)) is also concave by the LiebRuskai theorem that expresses the positivity of the metric given by the Hessian
of this T race. (See [21] for the computation of curvatures of such metrics.)
55
Is there a ”natural” (symplectic/Kählerian?) definition of these metrics that
reveals their geometry and makes transparent the entropy inequalities?
(b) Do the Euclidean Brascamp-Lieb inequalities remain valid for the quantum entropy of operators (say, with continuous compactly supported kernels
K(x1 , x2 )) on the L2 -Hilbert space L2 (Rn , dx) (for the Lebesgue and/or for the
Gauss measure dx on Rn ) with respect to the reductions in the tensorial decompositions L2 (Rn ) = L2 (X1 ) ⊗ L2L
(X2 ) for the (not necessarily coordinate)
orthogonal decompositions Rn = X1 X2 ?
5.10
Reduction Inequality for Orderable Groups Γ.
Let Γ = ×i Γi , i ∈ I, take a linear space B of F-functions on Γ with finite
supports for some field F and recall the asymptotic ΓJ \ -dimensions |D : ΓJ \ |
(see 1.10) for ΓJ \ = ×i∈I\J Γi . By combining the Shearer-Loomis-Whitney
projection inequality (P◦ ) with the equivalence ◦ ⇔ ∗ with order we conclude
that
If Γ is a left orderable amenable group, then
(P∗α )
rank(B) ≤
Y
|D : ΓJ \ |αJ
J⊂I
for all partitions of unity {αJ } on I and all linear (sub)spaces B of functions
on Γ with finite supports.
Furtermore, if the product map ×i Γi → Γ, for amenable subgroups in an
orderable (not necessarily amenable) Γ is one-to-one, then
(P∗ )
Y
(rank(D))k−1 ≤
|D : Γi |
i=1,2,...,k
for all linear spaces D of functions on Γ with finite supports.
Remarks. (a) The above generalizes to other types of “expanding” families of
subgroups, e.g. in Abelian Γ with the use of the corresponding Brascamp-Lieb
kind of inequalities. These, however, are not so good as far as the Brascamp-Lieb
constant l is concerned, albeit the discrete version, delivered by the Shannon
Expansion Inequality, is sharp, for the trivial reason; the presence of the δmeasures. A true generalization/linearization of the Brascamp-Lieb inequality
for (discrete) groups Γ and their group algebras A needs additional invariants
of measures in Γ and/or linear subspaces in A that would rule out their concentration on small subsets.
(b) If ΓZI , then (P∗ ) applies to linear subspaces D in the algebra of regular
functions on Zariski open subsets in FI , where the J-reduction of D becomes
the restriction DJ of (the functions in) D to a generic J-plane in Fk , where
|DJ | = |D : ΓJ \ | and the so interpreted (P∗ ) makes sense and remains valid for
the algebra of formal (and convergent, for normed fields) power series at a point
in FI . Furthermore, there are similar inequalities for the restrictions of D to
generic members of non-plane families of algebraic (analytic) subvarieties in FI .
56
5.11
•-Convexity and the Entropic Følner Functions F• .
A positive monotone increasing function F (n), n > n0 > 0, is called •-convex
if logF ( 1ε ), 0 < ε ≤ 1/n0 , is convex, that is equivalent to the convexity of
J(exp(l)), where J1 is the inverse function of F .
Examples. The functions const · nk are •-convex for all k ≥ 1 and const > 0.
Also the functions const · exp(na ) are •-convex for a ≥ 0.
Every positive monotone increasing function F (n) admits a unique maximal
•-convex minorant F• (n) ≤ F (n) and this F• , as it is easy to see, has growth
(roughly) comparable to that of F ,
F• (n) ≥ min max(αF (n), F (n)1−α )
α
for 0 ≤ α ≤ 1.
Given a positive function (”isoperimetric profile”) I(r), r ≥ 0 define the
associated
entropic Følner function, denoted F• (n) as the above maximal •-convex
minorant of the ordinary Følner function F (n) = FI (n). Clearly, the inverse
function to F• equals the minimal monotone increasing majorant M (r) of r/I(r)
1
for which M (exp(l))
is convex.
If X is a Γ-space with the orbit partition P and the ordinary Følner function
F then F• minorizes (not only the cardinalities of the subsets Y ⊂ X with
|∂(Y )| ≤ n, but also) the entropy of the partition P |Y , of Y :
If |∂(Y )|/|Y | ≤ 1/n, then
X
ent(P |Y ) =def |Y |−1
rs log(rs ) ≥ log(F• (n))
s∈P |Y
for rs = |s ∩ Y |, where the sum is taken over the P -slices s of P |Y
Proof. The boundary of Y is bounded from below, according to the definition
of I(r), by
X rs
X
P
I(rs )/|Y | =
J(rs )
|∂Y | ≥
s rs
s
s
for J(r) = I(r)/r. If J(exp(l)) is convex, then
P
X rs
rs log(rs )
P
J(rs ) ≥ J exp s P
= J(exp(ent(P |Y )))
s rs
s rs
s
and the proof follows from the definition of the Følner function F = F◦ of the
action.
P
The above equally applies to general partitions P (with s I(rs )/|Y | taken
for the definition of the “boundary”) and shows that the Følner transform of
•-convex functions for families of partitions Pi of an X is governed by the lower
bounds on the entropy of subsets Y ( recall that ent(Y ) =def log|Y | for discrete
spaces) by the entropies of the partitions. In particular, if
X
κi ent(Pi |Y ) + log(β)
ent(Y ) ≥
i
for all Y ⊂ X, then the Følner transform F = F (ni ) of •-convex functions Fi
satisfies
57
F (ni ) ≥ βFi (ni )κi .
Thus, for example,
the Shannon inequalities yield (often sharp) lower bounds on the multivariable entropic Følner functions of Cartesian products X = ×i Xi of Γi -spaces in
terms of the entropic Følner functions of individual Xi and/or of XJ = ×i∈J Xi ,
J ⊂ I.
This applies to the ordinary Følner functions if these are •-convex or (well)
approximated by •-convex functions. The simplest instance of this is the (multivariable) isoperimetric inequalities for free Abelian groups where the (ordinary)
Følner functions are •-convex.
Similarly,
the Shannon inequalities provide lower bounds on the Følner functions of
discrete groups Γ with families of subgroups Γi , i = 1, 2, ..., k, where the product
map ×i Γi → Γ is injective
and
the Brascamp-Lieb inequalities give similar bounds on the multivariable Følner
function of Γ = Rk with a given family of linear subspaces Γi ⊂ Γ, where one
may use the Følner functions Fi defined with the ordinary Euclidean boundaries
of subsets in Γi .
By combining the above with the Schwartz symmetrization for normal extensions,
1 −→ Γ1 −→ Γ −→ Γ2 −→ 1
we conclude that
the entropic Følner function of Γ is bounded from below by the product of
these of Γi , i = 1, 2,.
Finally, we recall ◦ ⇔ ∗ (see 3.2) and obtain a similar inequality for the
linear algebraic F∗ (Γ) for the space of functions with finite supports on Γ,
If the group Γ2 is orderable and the Følner functions F∗ (n1 , Γ1 ) and F◦ (n2 , Γ2 )
are minorized by •-convex functions Fi (ni ), i = 1, 2, then
F∗ (n1 , n2 ; Γ) ≥ F1 · F2 .
5.12
Entropic Poincaré Inequalities.
Consider a measure µ = f (x)λ on a Γ-space X, where λ is an invariant measure
on X, e.g. X = Γ with the Haar measure, and let Yµ = {f (x) ≤ m} ⊂ X × R+
be the subgraph of f .
We define |∂(µ)| = |∂(Yµ )| for the obvious action of Γ on X × R+ (for
whichever notion of the boundary for subsets) and then apply the above entropic
inequality to the partition of Yµ to the levels f (x) = m, m ∈ R+ . Since the
entropy of this partition is ≤ ent(µ) by the Shannon inequality, we see that
if the Følner function of X is minorized by a •-convex function F (n), then
ent(µ) ≥ F (|µ|/|∂(µ)|)
58
for all finite measures
µ on X (where, recall, |µ| =def µ(X) = |Yµ | and ent(µ) =
R
log|µ| − |µ|−1 S log f1 dx for the support S ⊂ X of µ.
Remarks.(a) The above log-Poincaré (Sobolev) inequality in the Rk case
holds, in the sharp form, for the Euclidean boundary, (see [12]).
(b) Measures µ on X, rather than subsets Y ⊂ X, make a natural domain of
definition for the Følner transform and the entropic Følner functions. In fact,
everything in the previous section (trivially) generalizes to measures, where
some of the resulting log-Poincaré inequalities become sharp in this context.
For example, the extremal measures for the log-Poincare inequalities associated
to the Euclidean Brascamp-Lieb inequalities are the Gaussian ones. (See [12]
and references therein).
6
Lower Bounds on F∗ for Groups with Large
Orderable Subgroups.
We bound from below the linear algebraic algebraic Følner functions of groups Γ
by applying the projection inequalities to certain infinitely generated orderable
(e.g. free Abelian) subgroups in Γ.
6.1
Expanding Subgroups Inequality.
Let Γ be a group with a finite generating subset G ⊂ Γ and let N = N (n) denote
the maximal number of elements gi ⊂ G·n ⊂ Γ, such that
(a) the subgroup ΓN , N = N (n), generated by gi , i = 1, 2, ..., N , is orderable;
(b) the product map of the (infinite cyclic) subgroups Γi generated by gi to
Γ, that is ×i=1,2,...,N (n) Γi → Γ is injective.
Then
F∗ (n; Γ, G) ≥
1 N( n )
2 2 .
2
Proof. Let µ be the probability measure uniformly distributed over g1 , ..., gN (n) .
Then the Loomis-Whitney inequality for expanding partitions implies that all
finite subsets Y ⊂ ΓN with
|∂µ |(Y ) ≤
n
1
|Y |
2
have |Y | ≥ 2 2 , and the proof follows with G∗m (see 1.4 and 1.6).
This can be modified and generalized in a variety of directions. For example,
there is a similar inequality for groups Γ, where some orderable subgroups ΓN ⊂
Γ admit N -long normal series, where the linearized isoperimetry is conveniently
done with F• -functions.
However, these inequalities do not give much new compared to what follows
by other means.
59
6.2
Distorted and Undistorted Actions on l2 and on Decaying p-adic Functions.
Consider an action of the group Γ = Zk on a linear space L over a field F and
show that the following properties
of such an action are equivalent
P
1. The action is free: if γ cγ (l) = 0, then either all cγ ∈ F or l ∈ L are
zero.
2. The action is non-degenerate: |D : Γ| > 0 for all non-zero subspaces
D ⊂ L i.e. if Hi ⊂ Γ is an exhaustion of Γ by Følner sets then
lim sup |Hi ∗ D|/|Hi | > 0.
i→∞
3. The action is uniformly non-degenerate:
infD lim inf |Hi ∗ D|/|Hi | > 0.
i→∞
4. The spans of these Hi -orbits satisfy
lim sup |Hi ∗ D|/|Hi |
k−1
k
= ∞.
i→∞
5. The action of the group F-algebra A of Γ on L does not factor through
an algebra of transcendence degree < k.
To see the equivalence of these conditions, observe that every finitely generated module L over A equals the set of F-sections of a coherent sheaf L over the
×
torus (F )k , where obviously, 1–5 are satisfied if and only if L is not supported
on a proper subvariety in (F× )k .
Remark. It is clear that 1 ⇒ 3 ⇒ 2 for all amenable groups Γ; furthermore, if
the group algebra A of Γ contains no non-constant finite dimensional subalgebras
(e.g. A has no zero divisors) then also 2 ⇒ 1; but it is unclear what corresponds
to 4 and 5 in the general case.
Examples: Functions with Fast Decay. (a) The action of Γ = Zk on the
Hilbert space l2 (Γ) satisfies the conditions 1–5. and the same remains true for
the space l2 (X), where X is a set with a free action of Γ by the following theorem
of Gabor Elek (see [8]).
If the group algebra of an amenable group Γ over R has no zero divisors,
then the action of Γ on l2 (Γ) is free.
Alternative Proof. Regard C-valued (moderately growing) functions a on Zk
as Fourier transforms of distributions â on the torus Tk : if this distribution has
a Zariski dense support then the linear span L of a satisfies 1-5 as we just seen.
In particular, this proves 1–5 for l2 (Zk ) since the â ∈ L2 (Tk ) for a ∈ l2 (Zk and,
being a measurable function, â has Zariski dense support.
(b) The spaces of decaying functions on Zk with values in discrete valuation
fields also satisfy 1-5, since they are representable by power series.
(c) Counterexample for Decaying C-valued Functions. The Fourier transform
a of a distribution â supported on an algebraic subvariety V ⊂ Tk of positive
codimension is annihilated by a polynomial in the group algebra of Zk that
vanishes on V with a sufficient multiplicity; thus (a) is violated. But such an
a = a(ni ) may be a decaying function on Zk .
For instance, let V ⊂ Tk be a smooth subvariety in that is nowhere infinitesimally flat (if V is analytic this means it is not contained in a sub-torus of Tk )
60
and â be given by a smooth measure on V . Then a does decay, being expressed
by an oscillating integral on V .
P
1−n
The fastest rate of decay, of such an a is O( i=1,2,..,k |ni |) 2 : smooth hypersurfaces V ⊂ Tk with non-vanishing Gaussian curvature, e.g. round spheres,
give such decay. (This was pointed out to me by Terence Tao.)
P
1−n
Question. Does a faster rate of decay, i.e. o( i=1,2,..,k |ni |) 2 , rules out
distributions with Zariski non-dense supports?
Domination Lemma.Let L be a space of functions on Γ = Zk that is
finitely generated as a Zk -module and that satisfies 1–5. Then the isoperimetry
of L dominates (as defined in 3.2) that of Γ. Consequently, the linear spaces of
the real l2 -functions and the fast decaying p-adic functions on Γ have the same
isoperimetry as Γ itself.
Proof. The linear space in question equivariantly embeds into the space of
formal power series and the Domination by Ordering from 3.2 applies.
In particular, it follows from the lemma that the combinatorial projection
inequalities for families of partitions of Zk , such as the Loomis-Whitney and
Shearer inequalities, pass to the linear subspaces D in these spaces L of functions, where the cardinality of the image of the projection Y → Zk /Γi for a
(free Abelian) subgroup Γi ⊂ Zk is replaced by the dimension of the span of the
Γi -orbit of D over the fraction field of the group ring of Γi .
Here are two applications.
1. Split Abelian Subgroup Inequality. Let Γ be a group with a finite
generating subset G ⊂ Γ and let N = N (n) denote the maximal number of
elements gi ⊂ G·n ⊂ Γ, such that gi , i = 1, 2, ..., N (n), generate a free Abelian
subgroup of rank N (n). Then the max-Følner functions F∗ of l2 (Γ) and of the
spaces of decaying functions with values in ultrametric fields satisfy
F∗ (n; Γ, G) ≥
1 N( n )
2 2 .
2
Proof. Use the measure µ uniformly distributed on gi and proceed as in the
above Expanding Subgroups Inequality.
2. Distorted Abelian Subgroups Inequality. Let Γ contain a finitely
generated free Abelian subgroup Γ0 and let G ⊂ Γ be a generating subset containing the generators of Γ0 . Then the max-Følner functions F∗ for the spaces
of l2 -functions and of decaying “ultrametric” functions on Γ are bounded from
below by the relative growth function G◦◦ (n) = |Γ0 ∩ G·n | as follows,
F∗ (n; Γ, G) ≥
1
n
G∗∗ ( ).
2
2
Proof. Use the measure µ uniformly distributed on Γ0 ∩ G◦n and apply
the linearized Displacement Inequality from with the G∗n -inequality as in the
Coulhon-Saloff-Coste argument in 1.4.
These 1 and 2 can be applied, e.g. to solvable groups as these have many
Abelian subgroups.
Example. Every non-virtually nilpotent polycyclic group contains an Abelian
subgroup with exponentially growing G◦◦ ( this is well known and easy); therefore
61
the l2 and ”ultrametric” Følner functions F∗ (n) grow exponentially for all
non-virtually nilpotent polycyclic groups.
Questions Can one replace “polycyclic” by “solvable without torsion” with
a suitable class of µ generalizing those in 1 and 2?
Is the ”free Abelian” requirement truly necessary in 1 and 2 or something
like ”orderable” would suffice?
Remark. The linearized L(oomis)-W(hitney)-S(hearer) inequality, that is an
upper bound on the dimension of a linear space D, say, of (germs of) analytic
functions on Rk (or any other class of functions with similar genericity properties) by the dimensions of the restrictions of D to generic members of parallel
families of the coordinate planes, can be proven by induction on k and dim(D)
as follows.
Take a generic hyperplane H ⊂ Rk parallel to the coordinate hyperplane
k−1
R
⊂ Rk , let D|H be the restriction of D to H and D/H ⊂ D be the subspace of functions that vanish on H. Then the LWS inequalities for D|H and
Q
Q
1
1
D/H yield that for D by the Minkovski inequality ( j (aj + bj )) l ≥ ( j aj ) l +
Q
1
( j bj ) l . (There are similar inequalities for non-coordinate ”webs of subvarieties” that we shall not discuss here.)
6.3
Isoperimetry with ”Large” H ⊂ Γ.
The above suggest regarding (various) boundaries |∂|H (Y ) as something depending on the size of H as well as of Y .
Here are two examples for Γ = Z and a subset H ⊂ Z.
(1) Let µ be the probability measure uniformly distributed on H. Then all
finite subsets Y ⊂ Z satisfy
(∂µ min).
|∂µ (Y )| = def
1 X
1
1
|∂h (Y )| ≥ min
|Y |, |H|
H
2
8
h∈H
To show this, consider two cases.
Case 1. Let |H| ≥ 2|Y |. Then the proof follows from the Displacement
inequality in 1.4.
Case 2. Divide H into positive and negative parts, H = H+ ∪ H− , and let
Y− ⊂ |Y | be the first (for the natural order) k− elements in Y for k− = |H− |
and Y+ ⊂ |Y | be the last k+ = |H+ | elements. Clearly,
|∂H+ (Y+ )| ≤ |∂H+ (Y )| and |∂H− (Y− )| ≤ |∂H− (Y )|
and the proof reduces to Case 1.
Remarks. (a) The inequality (∂µ min) yields a lower bound on the maxprofile of Z with respect to H, but the exact evaluation of I◦max (r; Z, H), even
for H = {1, 2, ..., k}, is not apparent and seems to pertain to the combinatorial
number theory.
(b) The inequality (∂µ min), obviously, remains true for all biorderable
groups Γ.
(c) Let Hi ⊂ Γ, i = 1, 2, ...k, be subsets, such that the product map
(h1 , h2 , ..., hk ) 7→ h1 · h2 · ... · hk of the Cartesian product H1 × H2 × ... × Hk to
Γ is one-to-one. Then, by the G·m -inequality,
62
X
|∂µi (Y )| ≥ |∂µ (Y )|,
i
where µi are the uniform probability measures on HiQand µ is such measure on
the product H = H1 · H2 · ... · Hk ⊂ Γ. Since |H| =P i |Hi | under the ”one-toone” assumption, one gets a good lower bound on i |∂µi (Y )| for biorderable
(and sometimes more general) groups Γ.
(2) If H ⊂ Z admits no nontrivial relation
X
nh · h = 0
h∈H
where |nh | ≤ N for a given N ≥ 0 and all h ∈ H, then every subset Y ⊂ Z of
cardinality |Y | ≤ N satisfies
Y
|Y |k−1 ≤
|∂h (Y )|.
h∈H
Proof. Let Ci be the (Cayley) graph on the vertex set Y , where the edges
are the pairs (y, y + hi ) ∈ Y × Y (for the group composition in Z depicted by
”+”).
The number of the connected component in Ci , that is the number li of
the maximal arithmetic hi -progressions in Y , equals |∂hi (Y )|. Furthermore, the
family of the k partitions Pi of Y into the components of Ci is expanding
Qin the
sense of 5.7 due to the absence of the above relations. Hence, |Y |k−1 ≤ i li by
the Loomis-Whitney inequality.
Q
(In general, the inequality |Y |k−1 > i li implies, for given positive integers
Ni , either the existence, for somePi, of an arithmetic hi -progression in Y of
0 with |ni | < Ni .)
length Ni , or a nontrivial relation i ni hi =P
Example If H = {hi }, where |hi+1 | > N j≤i |hj |, then the ” no relation”
condition is obviously satisfied.
The above easily generalizes to the sparse subsets H in all Γ, where one may
have a much larger |∂H | with relatively small H for non-Abelian groups Γ.
On the other hand, the exact evaluation of the max-profile is not apparent
already for Γ = Z and H = {1, 2, ..., k} (where the problem pertains to the
combinatorial number theory.)
This suggests
Dual Isoperimetric Problem: Find, for every r = 1, 2, ... and 0 ≤ α ≤ 1, a
”minimal” subset H ⊂ Γ, such that
|∂|max
H (Y )
≥α
|Y |
for all Y ⊂ Γ with |Y | ≤ r , where ”minimal”, may refer to the cardinality of
Y , or to the radius of the minimal ball containing H, or to another notion of a
size of a subset in Γ.
It seems that most (all?) proofs of lower bounds on the isoperimetric profiles
of groups Γ also deliver, under a close inspection, ”small” H giving ”large” maxboundaries to all Y ⊂ Γ.
63
7
Linear Algebraic Entropic Inequalities for Normal Group Extensions.
We prove in this section coarse product inequalities for a suitably defined ”filtered” (compare 1.11) entropic Følner functions of linear actions that apply, for
example, to pure torsion groups where other techniques are not available.
7.1
Vertical Families L of Linear Subspaces and L-Entropy.
Let L be a linear space over some field with a distinguished class L of what we
call vertical linear subspaces in L, where we assume this class to be closed under
finite and infinite intersections. If D ⊂ L is a linear subspace, then we denote
by L|D the induced class of subspaces in D that are the intersection of D with
the subspaces from L.
Tensor Product Example. Consider linear spaces Lhor and Lvert and take
L = Lhor ⊗ Lvert , where L consists of the subspaces L0hor ⊗ Lvert for all linear
subspaces L0hor ⊂ Lhor . Clearly, L consists of all subspaces invariant under the
group of linear transformations of Lvert naturally acting on L.
Let L equals some Γ-invariant space of functions on a
Γvert -Examples.
group Γ and L consists of all Γvert -invariant subspaces in L for some normal
subgroup Γvert ⊂ Γ.
The quotient group Γhor does not, in general, act on L but it does act on L
by projective transformations where each individual transformation comes from
a linear transformation of L.
This class L is not always closed under complements: if L1 ⊂ L2 for L1 , L2 , ∈
L, then the quotient space L2 /L1 is not always Γ0 -equivariantly isomorphic to
anybody in L.
Here are two particular classes L where the complement always exists.
1. Support Saturated class. A subspace L1 ⊂ L is called support saturated if
it contains all functions from L which have supports contained in the support
of L1 .
If L consists of the functions with finite supports in X, then
the class of all Γvert -invariant support saturated subspaces admits complements corresponding to taking complements of Γvert invariant subsets in X.
2. l2 -Spaces. If the underlying field is R or C and L = l2 (X), then
the class of all closed Γvert -invariant subspaces also admit (now just orthogonal) complements.
L-Length and L-Entropy. Take a finite dimensional linear subspace D ⊂ L
and consider all its L-”partitions“, i.e. decompositions into sums (spans) of
non-zero independent subspaces D = D1 ⊕ D2 ⊕ ... ⊕ Di ⊕ ... ⊕ DN , such that
the associated filtrations are L-vertical:
D1 ⊕ D2 ⊕ ... ⊕ Di ∈ L|D for all i = 1, 2, ...N.
Define the L-length of D as the maximal N , for which such ”partition”
exists and the L-entropy entL (D) as the supremum of the entropies of the finite
measure spaces made of N atoms of weights di = |Di | = rank(Di ) over all
L-”partitions” of D for all N = 1, 2, ....
Clearly,
64
|D| ≥ lengthL (D) ≥ exp(entL (D))
for all L and D.
Example. If L consists of all linear subspaces in L then L-length of D equals
|D| and entL (D) = log|D| but if L = {L0vert ⊗ Lhor } for an infinite dimensional
space Lvert ⊃ L0vert then generic finite dimensional subspaces D ⊂ L = L1 ⊗ L2
have entL (D) = 0.
L-Boundary. Given a finite dimensional linear subspace, define the g|Lboundary of D for a linear transformation g of L preserving L by
|∂g|L |(D) = supL0 ∈L |D ∩ L0 /g −1 (D)|.
Clearly, this equals the ordinary boundary |∂g (D)| if D ∈ L; in general,
|∂g|L |(D) ≤ |∂g (D)|.
Logarithmic Filtration Lemma. Let G be a set of transformations of L,
such that L is G-invariant, yet it contains no non-zero G-invariant subspace. If
a finite dimensional subspace D is contained in some L0D ∈ L and
|∂g|L |(D) ≤ ε|D|,
for all g ∈ G, then there is a descending sequence of subspaces D = D0 ⊃ D1 ⊃
D2 ⊃ ... ⊃ Di ⊃ ... ⊃ 0, where each Di = D ∩ L0i for L0i ∈ L and such that
(2 + ε)
|D/Di | ≤ 2|D/Di−1 | + ε
for all i = 1, 2, ....
Proof. Take a sequence γi of invertible linear transformations of L, where
γ0 = id, where γi ·γ −1 ∈ G for all i and where every transformation in the (linear)
transformation group generated by G appears as some γi in the sequence. Then
the sequence
Di0 = D ∩ γ0 (L0D ) ∩ γ1 (L0D ), ..., γi (L0 ).
Remark. If L is associated with a normal subgroup Γvert in Γ generated by G,
then the inequality (2 + ε) looks abysmally weak compared to the corresponding
(equally obvious) combinatorial intersection inequality (compare 1.4)
(ε)
|Y \ Yi | ≤ ε
where Yi = Y ∩ Yi0 for Γvert -invariant subsets in Γ. It is unclear to me if/when
2 can be removed from (2 + ε) for linear actions.
1
4 -Entropy Corollary. If
|∂|max
G|L (D) =def maxg∈G (D) ≤ ε|D|
for ε ≤ 14 , then
(ent ≥ 14 )
entL (D) ≥
65
1
.
4
Remark. The corresponding (also obvious) bound on the length of D,
lengthL (D) ≥ log3 (ε) − 2
looks somewhat better, but (ent ≥ 14 ), albeit independent of ε, better behaves
under induction as we shall see below.
7.2
L-Entropic Følner Functions.
Given a countable group with a (usually generating) subset G ⊂ Γ, consider all
countable overgroups Γ̃ → Γ → 1 with the kernel denoted Γvert ⊂ Γ̃ and let L̃
stands either for the space of F-valued functions on Γ̃ with finite supports on Γ̃
for a given field F, or for the space of the complex l2 -functions on Γ̃.
Denote by L̃ the family of Γvert -invariant spaces L̃0 of functions on Γ̃ with
either finite projections of their supports to Γ, or of l2 -spaces with finite von
Neumann Γvert -dimensions.
Define the internal G|L-boundary of a D̃ ⊂ L̃ by
|∂G̃|L |(D) = sup (|D ∩ L0 ∩g∈G̃ γ −1 (D)|,
L0 ∈L
where G̃ denotes some lift of G to Γ̃ and where, observe, the definition is indemax
pendent of a lift and |∂G̃|L |(D) ≥ |∂G̃|L
|(D).
Define the entropic Følner function F̃• (n) = F̃• (n; Γ, G) as the maximal
•-convex function, such that
exp(entL (D̃)) ≥ F̃• (n)
for all finite dimensional subspaces D̃ ⊂ L̃ satisfying
|∂G̃|L̃ |(D) ≤
1
|D|.
n
for all overgroups Γ̃ and all finite dimensional subspaces D̃ ∈ L̃.
If there are several Gi , let
F̃•maxi (n) = F̃• (ni ; Γ, Gi )
be the maximal •-convex function, such that
exp(entl (D̃)) ≥ F̃• (ni )
for all finite dimensional subspaces D̃ ⊂ L̃ satisfying
|∂G̃i |L̃ |(D) ≤
1
|D|.
ni
for all overgroups Γ̃ and all finite dimensional subspaces D̃ ∈ L̃, where the •convexity is understood as the convexity of log(F (1/εi )) as a function of the
vector-variable {εi }.
Super-multiplicativity Lemma. Let
1 → Γ1 → Γ → Γ2 → 1
66
and let G1 = G ∩ Γ1 and G2 ⊂ Γ2 be the projection of a given G ∈ Γ to Γ2 .
Then
max1,2
F̃•
(n1 , n2 ; Γ, G1 , G2 ) ≥ F̃• (n1 ; Γ1 , G1 ) · F̃• (n2 ; Γ2 , G2 ).
Proof. The L̃-”partition” of a given D ⊂ L̃ is constructed in two steps.
First we let L̃1 ⊂ L̃ be the the subclass of subspaces that are invariant
under Γ1 , or rather under the pullback of Γ1 to Γ̃ and take the L̃1 -”partition”,
D = ⊕i Di of D of the maximal entropy that is bounded from below with the
second Følner function by log(F̃• (n; Γ2 , G2 )).
Then we partition each Di with L . Since the entropy is additive (see
section 5) and the boundary ∂G|L is ”slice-wise”-additive the resulting entropy
is bounded from below by the sum of the log’s of the first and the second Følner
functions by the •-convexity of the first one as in 5.11.
Remark. This argument, unlike the one in 5.11, does not use the Shannon
inequality.
By combining this Lemma with the above 41 -Corollary, we arrive at the
following.
Chain Extension Inequality . Let 1 = Γ0 ⊂ Γ1 ⊂ ... ⊂ Γi ⊂ ... ⊂ ΓN ⊂ Γ
be a normal sequence, i.e. each Γi is normal in Γi+1 , and let Gi ⊂ Γi , be
subsets, such that the image of Gi in Γi /Γi−1 generates an infinite subgroup for
all i = 1, 2, .... Then
F̃•maxi (ni ; Γ, Gi ) ≥ exp(
N
)
4
for ni ≥ 4.
Since the F̃ minorizes the ordinary linear algebraic Følner function F∗ (for
functions with bounded supports and l2 -functions respectively), we conclude as
in 6.1, by applying the above to Cartesian products of infinite groups, that
the l2 -Følner function Fl2 as well as F∗ for finitely supported functions with
values in an arbitrary field of the wreath product Γ of Γ1 and Γ2 are bounded
from below in terms of the growth function of Γ2 by
1
Fl2 (n), F∗ (n) exp( G◦ (n; Γ2 )),
4
provided the group Γ1 is infinite (finitely generated).
There are amenable pure torsion groups, e.g. the wreath products of Grign
orchuk groups, with growth G◦ (n) ∼ 2n and where F∗ (n) and Fl2 (n) are 22 .
Questions. Do the higher iterates of wreath products of the Grigorchuk
groups have faster growing F∗ and Fl2 . Are there amenable groups (torsion or
n
no torsion) with Fl2 (n) growing faster than 22 ? What are the Følner functions
of the iterated wreath products acting on the spaces of decaying functions with
values in ultrametric fields?
(If Γ1 contains an element of infinite order, then the Følner function of
2
Γ = ΓoΓ
for decaying functions grows ε(1 + ε)G◦ (n;Γ2 ) , ε > 0.1 by 6.1).
1
Application to Grigorchuk Groups. If a finitely generated group Γ contains
a subgroup commensurable to Γ × Γ (e.g. a suitable Grigorchuk group), then,
by the super-multiplicativity,
67
F̃ (n; Γ) ≥ εΓ · (F̃ (εΓ n; Γ))2
for some strictly positive εΓ . It follows that
the Følner functions F∗ and Fl2 of such Γ grow at least as fast as exp(nα )
for some α = αΓ > 0.
Isoperimetry in Vertical Grassmannians. Let us regard a class L as a
subspace in the Grassmannian Gr(L) of linear subspaces L0 of a linear space L.
The notion of isoperimetry makes sense for a projective action of a group Γ on
L, whenever there is some notion of ”dimension” or ”rank” for linear subspaces
L0 ∈ L.
Such a notion is provided in the l2 -enviroment by the von Neumann Γvert dimension, when all L0 ∈ L are acted upon by a group Γvert as in the above
examples. Futhermore, if Γvert is amenable, there is the asymptotic rank defined
in 1.10.
It seems, such projective actions on sub-Grassmannians constitute a correct general setting for the ”linearized isoperimetry”. For example, the above
entropic argument applies here and delivers the corresponding isoperimetric inequalities. But I have not looked into the subject matter beyond this point.
8
Examples of Group Algebras with Fast and
with Slow Growing Følner functions.
We start with a bound on the isoperimetric profiles and Følner functions of
certain group algebras and then construct groups where the linear algebraic
profiles grow much slower than the combinatorial ones. Also we construct groups
with arbitrarily fast growing Følner functions of their group algebras.
8.1
An Upper Bound on I∗ for Split Extensions.
Consider an exact sequence of groups
1 → Θ → Γ → Γ0 → 1,
where Γ (and, hence, Γ0 ) is finitely generated.
If the sequence splits by some embedding Γ0 → Γ and the group Θ is locally
finite (i.e. every finitely generated subgroup in Θ is finite) then the isoperimetric
profile I∗ (r; A) of the group algebra A of Γ is equivalent to the profile I∗ (r; A0 )
of the group algebra A0 of Γ0 with an arbitrary coefficient field F. In particular,
if Γ0 = Z, then I∗ (r; A) is bounded.
Proof. Choose a symmetric finite generating set G = G0 ∪ G1 ⊂ Γ, where
G0 ⊂ Γ0 ⊂ Γ and G1 ⊂ Θ. Take a linear space D0 ⊂ A0 of F-valued functions
on Γ0 with supports in a finite subset Y0 ⊂ Γ0 . Denote by Θ1 ⊂ Θ the (finite!)
subgroup generated by yg1 y −1 for all y ∈ Y0 and g1 ∈ G1 .
Let a1 ∈ A be an F-valued function on Γ that equals 1 on Θ1 ⊂ Θ ⊂ Γ and
that vanishes everywhere else on Γ. Observe that a1 is invariant under all above
yg1 y −1 .
The embedding Γ0 → Γ makes Γ0 and hence its group algebra A0 act on
functions on Γ; we denote by D the linear span of the functions d ∗ a1 on Γ for
68
all d ∈ D0 where ”∗” stands for the (convolution) product in the group algebra
A of Γ.
Each g1 ∈ G1 ⊂ Θ trivially acts on y ∗ a1 for all y, y −1 ∈ Y0 . Indeed,
g1 = yθ1 y −1 for θ1 = y −1 g1 y ∈ Θ1 and y ∈ Y0 . Then
g1 ∗ (y ∗ a1 ) = (yθ1 y −1 ) ∗ (y ∗ a1 ) = (yθ1 ) ∗ a1 = y ∗ a1 ,
where θ1 , since it is contained in Θ1 , acts trivially on a1 . Therefore, the space
D is invariant, actually fixed, under the action of all g1 ⊂ G1 , because it is
contained in the span of the y ∗ a1 for y ∈ Y0 .
It follows that
G-boundary of D equals its G0 -boundary.
Since Γ0 freely acts on Γ and every Γ0 -orbit meets Θ ⊂ Γ at a single point,
the action of Γ on the span of the orbit of every non-zero function a on Γ with
support in Θ is isomorphic to the action of Γ0 on the orbit of the unit in its
group algebra that is the span of the orbit of the unit. This isomorphism sends
D → D0 and ∂G0 (D) → ∂G0 (D0 ); hence,
|D| = |D0 | and |∂G (D)| = |∂G0 (D)| = |∂G0 (D0 )|.
This shows that I∗ of Γ is bounded by I∗ of Γ0 , because D0 was an arbitrarily
chosen subspace in A0 .
The proof is concluded by observing that the reverse inequality follows from
(+)∗ in 1.5.
2
Examples. (1) Take the wreath product Γ = ΓoΓ
1 , where 2 ≥ |Γ1 | < ∞.
Then the combinatorial Følner function of Γ satisfies,
F◦ (n; Γ) (1 + α)F◦ (n;Γ2 )
for α > 0, while the Følner function of the group algebra of Γ is bounded by
F∗ (n; Γ) ≺ F◦ (n; Γ2 ).
Remark. The latter agrees with the bound
F◦ (n;Γ2 )
2
F∗ (n; ΓoΓ
,
1 ) ≺ F∗ (n; Γ2 )F∗ (n; Γ1 )
since F∗ (n; Γ1 ) ≡ 1 for finite groups Γ1 .
(2) There are groups Γ with two generators that are (necessarily split) extensions of locally finite groups Θ,
1 → Θ → Γ → Z → 1,
where one can have an arbitrary fast growing Følner function of Γ (see 8.2 below). Yet, by the above criterion, their groups algebras have bounded isoperimetric functions.
On Extensions of Finitely Generated Groups. If both groups Γ0 and Θ in
a (not necessarily split) exact sequence 1 → Θ → Γ → Γ0 → 1 are finitely
generated and infinite, then the max-isoperimetric profile of the group algebra
of Γ is unbounded; moreover
I∗max (r : Γ, G) log(r)
69
by the L-length inequality in 7.1.
Questions. Let an infinite finitely generated group Γ have I∗ bounded. Is it
then a Z-extension of a locally finite group?
Does any of the following conditions help?
(1) Γ is a torsion group;
(2) Γ contains an infinite finitely generated subgroup of infinite index;
(3) Γ contains an element of infinite order;
(4) Γ has no torsion.
If Γ has no torsion, does then the polynomial bound on the Følner function,
F∗ (r) rk for some k > 1, imply that Γ is virtually nilpotent?
8.2
Amenable Groups with Fast Growing Følner Functions.
Let FX denote the free group on a set X of generators and let [Y ]k ⊂ FX be
the set of the commutators [...[[x1 , x2 ], x3 ]...xk ] for all k-long sequences xi ∈ Y .
Denote by FX (p) the quotient group of Fx by the relation xp = 1 for some
prime number p and all x ∈ X (that is the free product of X copies of Zp ) and
observe that
1. If an element in FX (p) is representable by a word y1i1 ·y2i2 ·...·ykik ,where the
neighboring yi ⊂ Y are not equal and where no ij is divisible by p, then the image
of w under the composed (surjective) homomorphism h : FX → FY (p)/[Y ]k ,
FX → FY → FY (p) → FY (p)/[Y ]k
is 6= 1.
Here is the (standard)
Proof. Assume X = Y and let Wk be the Fp -linear space (formally) spanned
by the irreducible words w = y1j1 · y2j2 · ... · yljl ⊂ FX , where all j ≥ 0 and the
total degree of w satisfies deg(w) = j1 + j2 + ... + jl ≤ k.
Let Ay be the (nilpotent) linear operators on Lk defined by their values at
w ∈ Lk as follows, A(w) = yw for deg(w) < k and A(w) = 0 otherwise.
The map y 7→ 1 − Ay extends to a homomorphisms of FY (p)/[Y ]k to the
group of linear transformations of Wk , where each y −1 goes to the finite sum
1 + Ay + A2y + ... + Aky , since Ak+1
= 0. Then the non-equality h(w) 6= 1 follows
y
from non-vanishing of the product operator Ajy11 · Ajy22 · ... · Ayl jl .
This k-acyclicity/divergence of FY (p)/[Y ]k implies the following lower bound
on the Følner function of the (finite for p < ∞) group FY (p)/[Y ]k for the (joint,
see 1.1) boundary ∂Y , where we regard Y as a (generating) subset in FY (p)/[Y ]k ,
F◦ (n) ≥ ((|Y | − 1)(q − 1))αk
for q = min(n, p) and some α ≥ 1/4.
Remark. One can do better including estimates on the E◦ and P◦ functions
of the group N (k) = FY (p)/[Y ]k with respect to the cyclic subgroups generated
by y ∈ Y , while we shall need below only the (rough but uniform) bound,
(1k )
F◦ (n) = F◦ (n; N (k), Y ) ≥ k for all p ≥ 3 and |Y | ≥ 2
70
for all n ≥ 1, that is stated below in a form that is convenient for making our
examples.
2. Let k(Y ) be a function (rather than a number) on the subsets Y ⊂ X with
values 1, 2, 3...., ∞, such that Y1 ⊂ Y2 ⇒ k(Y1 ) ≤ k(Y2 ), and let k2 (Y )(≤ k(Y ))
be the maximum of k(Y 0 ) over all subsets Y 0 ⊂ Y of cardinality |Y 0 | = 2.
Let N (k) be the factor group of FX (p) obtained with the above (commutator)
relations [Yk(Y ) ] for all Y ⊂ Y , where [Yk(Y ) ] for k(Y ) = ∞ means ”no relation“.
Then
F◦ (n) = F◦ (n; N (k), Y ) ≥ k2 (Y )
for all Y ⊂ X with |Y | ≥ 2, all p ≥ 3 and n ≥ 1.
Next, let X be acted by a group Γ with a given finite generating set G ⊂ Γ
and let the function k(Y ) be Γ-invariant.
Then Γ acts by automorphisms on N (k) and thus, diagonally, on N (k) × X.
The group N (k) also acts on the product N (k) × X by translations on the first
factor and the group generated by the two actions is denoted by N (k) n Γ. If
the action of Γ on X is transitive, then the group N (k) n Γ is finitely generated,
namely, by the subset {x0 } ∪ G ⊂ N n Γ for an arbitrary x0 ∈ X ⊂ N n Γ,
3. Let DiamG (Y ) be the minimal number m, such that every point y1 ∈ Y ,
can be moved to any other point y2 ∈ Y by some γ ∈ G·m , i.e. γ(y1 ) = y2 , where
we assume that G is symmetric (under γ ←→ γ −1 ) and contains id. Denote by
k2 (m) the supremum of k2 (Y ) over all Y ⊂ X with DiamG (Y ) = m.
The Følner function of the action of N (k) n Γ, {x0 } on N (k) × X with the
generating subset {x0 } ∪ G ⊂ N (k) n Γ, {x0 }satisfies
F◦ (mn; N (k) × X) ≥ k2 (m)
for n ≥ 1 and all m = 2, 3, .... In particular, if k(Y ) depends only on m =
DiamG (Y ), then k2 = k and
F◦ (mn; N (k) × X) ≥ k(m).
Proof. Combine the above with (n/m) in 1.9.
This shows that one can make F (n) grow arbitrarily fast with a fast growing
function k(m) and, at the same time, have N (k) locally nilpotent, thus, N (k)nΓ
amenable, if Γ is amenable.
Here is the summary.
Example: Amenable Groups with F◦ ≥ κ(n). Let X = Γ be an
amenable group with a finite generating subset G ⊂ Γ and let κ(n), n=1,2,..., be
an arbitrary monotone increasing Z+ -valued function. Then the group N (k) n Γ
for k(Y ) = κ(Diam(Y )2 ), Y ⊂ Γ, is finitely generated amenable and
F◦ (n; N (k) n Γ, G) ≥ κ(n).
for all n ≥ 1.
Examples with the Linear Algebraic Følner functions F∗ . If p = ∞
then the group N (k) is orderable and the above yields
the lower bound
F∗ (n; N (k) n Γ) ≥ κ(n)
71
for the (action of N (k) n Γ on) functions with compact supports on N (k) n Γ
with values in an arbitrary field (where one does not need Γ to be orderable).
On the other hand,
if p ≤ ∞, then
F∗ (n; N (k) n Γ) ≤ F◦ (n; Γ)
while F◦ (n; N (k) n Γ) can grow as fast as one wishes.
Remark and Questions. (a) The existence of groups N n Z with arbitrarily
fast growing F◦ , where N is a locally finite group, is indicated in [9] and groups
of intermediate growth with arbitrarily fast growing F◦ are constructed in [10].
(b) It is not hard to give an upper bound on F◦ for the above group and
observe that
F∗ (n; N (k) n Γ) ∼ κ(n)
for the functions κ(n) with sufficiently fast growing derivatives.
(c) Can one achieve the above with a construction (invariant under all permutation of X) with k(Y ) depending on the cardinality, rather than the diameter
of Y ? (Possibly, this can be done with the existence theorem for non-Abelian
free subgroups in groups with universal commutator relations, [3]).
(d) Is there an universal bound on the asymptotic growth of the Følner
functions of finitely presented amenable groups by a recursive (primitively recursive?) function? (Maybe there is such a bound in every given recursive
class of presentations?). Or, at another extreme, are there finitely presented
amenable groups with so fast growing F◦ (n), such that their amenability is unprovable in Arithmetic? (An enticing possibility would be this situation for the
Thompson group).
9
E◦ -Functions and Isoperimetric Inequalities
for Wreath Powers of Partitions and Groups .
The isoperimetric inequalities in the previous section are similar to those for the
wreath product of groups that were obtained earlier in an asymptotically sharp
form in [9] with an (implicate) use of E◦ -functions. We shall present below a
translation of the argument from [9] to the language of partitions that provides
a quantitative (better constants) improvement of the isoperimetric inequalities
for iterated wreath products of groups and a qualitative (asymptotic behavior)
improvement for infinitely iterated wreath products.
9.1
Wreath Powers of Sets.
Let Z be a discrete set with a distinguished family S of subsets (slices) S ⊂ Z,
and define the o-space associated to S, denoted Zo , that is the set of of the pairs
(S, z) for S ∈ S and z ∈ S.
The simplest (and the largest) such Zo is obtained with S = 2Z , the family
of all subsets in Z.
The set Zo naturally carries two partitions P1 and P2 : the slices of P1 are
obtained by varying z in the pair (S, z) within S, while the slices of P2 consist
of the pairs (S, z) with fixed z and all S 3 z.
72
Basic example. Let Z = X I with S being the family of the ”coordinate
lines”. In this case we write Zo = Z ×I = X oI and denote the two corresponding
partitions of X oI by P1 = PX , where all slices are copies of X, and by P2 = PI ,
where the slices are copies of I.
Recall that this S is Shannon: the geometric mean of the cardinalities of the
S-slices of every Y ⊂ Z is bounded by the cardinality of Y ,
Y
1
|S ∩ Y | |Y | ≤ |Y |,
S
where the product is taken over the slices S ∈ S with non-empty intersections
S∩Y.
Evaluation of E◦ , P◦ and F for Zo . If S is Shannon, then
(E◦ )
2
E◦ (L1 , L2 ; P1 , P2 ) = L2 · LL
1 ,
(P◦ )
P◦ (n1 , n2 ; Zo /P1 , Zo /P2 ) = n2 · nn1 2
and
(F oF )
F (F1 (n1 ), F2 (n2 )) ≥ d2 F2 (ε2 n2 )(d1 F1 (ε1 n1 ))d2 F2 (ε2 n2 ) ,
where,
1 − ε1 − ε2
1 − εi
for arbitrary positive numbers εi , i = 1, 2 satisfying
di =
ε1 + ε2 < 1.
Furthermore, if F2 (n) = const · n and F1 (n) is •-convex, then
(F• ).
F (F1 (n1 ), F2 (n2 )) ≥ F2 (n2 )F1 (n1 )F2 (n2 )
Proof. If a subset Y ⊂ Zo has the P1 -slices of cardinalities ≥ L1 and the
P2 -slices of cardinalities ≥ L2 then its projection Y 0 to Z has at least L2 slices
of cardinalities ≥ L1 at each point in Z. Then, by the definition of ”Shannon”,
L2
2
|Y 0 | ≥ LL
1 and |Y | ≥ L2 · L1 . This proves (E◦ ) and, similarly, (P◦ ) follows
B
from the A -inequality (see 5.3, 5.7).
To prove (F oF ) we take a Y ⊂ Zo with
X
I1 (|S ∩ Y |) ≤
1
|Y |
n1
I2 (|S ∩ Y |) ≤
1
|Y |
n2
S∈P1
and
X
S∈P2
73
for the ”isoperimetric profiles” Ii associated with Fi and let Yi ⊂ Y , i =
1, 2, be the union of the Pi -slices of cardinalities ≥ Fi ((1 − εi )ni ). Then the
intersection Y• = Y1 ∩ Y2 is non-empty and has ”large average measures” of
P1 - and P2 -slices according to [∩i /Pi ] in 4.5. Then (AB ) applies with A =
d1 F1 (ε1 n1 )) and B = d2 F2 (ε2 n2 ).
Finally we turn to F• and observe that the inequality
X
I2 (|S ∩ Y |) ≤
S∈P2
1
|Y |
n2
implies that the number of the P2 -slices of Y is bounded by
|P2 ∩ Y | ≤
1
|Y |
n2
and the proof follows by the •-convexity of F◦ (n; Γ1 ).
Remarks. (a) The evaluation of the E◦ function for X oI (and for Zo associated to an expanding family of partitions Pi in general) can be seen directly by
counting Pi -chains in Z, (see, [9] where it is done, in the language of hypergraphs
rather than partitions, for wreath products of groups and graphs).
(b) There are further constructions and partitions associated to power spaces
X I , e.g the o-spaces associated to the ”plane” rather‘ than ”line” partitions.
Then, one can take Cartesian powers of Zo that are products of the original Pi .
Evaluation of the invariants the resulting families of partitions remains an open
problem.
(c) It is unclear, in general, if there is a Shannon type inequality for measures
µ on (rather than subsets in) Zo , except for the case of the atoms in each P2 slice S having equal weights, denoted µS , where the Shannon inequality relates
the entropy of P1 and the measure of P2 with the weights S 7→ µS ,
(?)
µ(Zo )
µ(Zo )
entµ (X) ≥ entµ (Zo ) = P
+ log P
.
µ
S
S∈P2
S∈P2 µS
But it is doubtful that the concepts of ”measure“ and/or ”entropy“ adequately reflect the combinatorics of such families of partitions Pi .
9.2
Følner Functions for Iterated Wreath Product of Groups.
Given groups Γ1 and Γ2 , denote (with a minor abuse of notation) by Γ1Γ2 the
restricted Cartesian power group consisting of the functions θ : Γ2 → Γ1 with
finite supports (i.e. equal id ∈ Γ1 away from a finite subset in Γ2 ) and observe
that both Γ1 and Γ2 naturally act on ΓΓ1 2 , where Γ1 is implemented in ΓΓ1 2 by
the functions trivial (i.e. = id ∈ Γ1 ) away from id ∈ Γ2 . The group generated
by these two actions is called the wreath product of Γ1 and Γ2 , that we denote by
Γ1oΓ2 . (The customary notation for the wreath product is Γ1 o Γ2 and/or Γ2 o Γ1 ).
There is a natural exact sequence
2
1 → ΓΓ1 2 → ΓoΓ
→ Γ2 → 1,
1
where ΓΓ1 2 is given the coordinate-wise group structure (as the Cartesian product
2
of Γ2 copies of Γ1 ) and where Γ1 is embedded into ΓΓ1 2 and, hence, into ΓoΓ
1 ,
74
by functions θ : Γ2 → Γ1 such that θ(γ2 ) = id ∈ Γ1 for γ2 6= id ∈ Γ2 . This
2
sequence is split by the obvious embedding Γ2 → ΓoΓ
1 .
oΓ2
Observe that the group Γ1 is finitely generated if Γ1 and Γ2 are finitely
2
generated, while the subgroup ΓΓ1 2 ⊂ ΓoΓ
is infinitely generated for infinite
1
groups Γ2 .
The wreath product is a binary operation on groups, now written Γ1 o Γ2 for
Γ1oΓ2 , that is neither commutative nor associative: the composed wreath product
of k groups is determined by the way we put the brackets. Here are three (most
symmetric) examples for k = 8,
.(.(.(...
Γ1 o (Γ2 o (Γ3 o (Γ4 o (Γ5 o (Γ6 o (Γ7 o Γ8 )))))),
.).).)...
((((((Γ1 o Γ2 ) o Γ3 ) o Γ4 ) o Γ5 ) o Γ6 ) o Γ7 ) o Γ8
and
...(()())...
((Γ1 o Γ2 ) o (Γ3 o Γ4 )) o ((Γ5 o Γ6 ) o (Γ7 o Γ8 ))
The E◦ -functions (obviously) follow the composition rule for partitioning
slices and since the E◦ -function of the simple wreath product equals the nu2
merical o-product, i.e. L1 o L2 =def L2 · LL
1 . the E◦ -functions of the iterated
wreath products relative to the subgroups Γi equal the corresponding composed
o-product with the same bracketing as for the corresponding Γ, e.g.
E(L1 , L2 , L3 ...) = L1 o (L2 (o(L3 o ....)))...
for .(.(.(...
Now we invoke the lower bound of the Følner functions by E◦ -functions (see
4.5) and obtain
Erschler Inequality for Wreath Products. The multivariable Følner
function of an iterated wreath product of Γi is bounded from below by the correspondingly composed o-product of ci Fi (εi ni ), where Fi are
Pthe Følner functions
of Γi and ci , εi are arbitrary positive constants satisfying i (εi +ci (1−εi )) < 1.
Remark. The above (F oF ) serves slightly better for the simple products,
F◦ (n1 , n2 ; Γ1oΓ2 ) ≥ d2 F2 (ε2 n)(d1 F1 (ε1 n))d2 F2 (ε2 n) ,
where,
di =
1 − ε1 − ε2
1 − εi
for arbitrary positive numbers εi , i = 1, 2 satisfying ε1 + ε2 < 1.
Similarly, (F• ) implies that
n2
F• (n1 , n2 ; ΓoZ
1 ) = n2 F• (n1 , Γ1 ) .
Question. Is there a sharp bound on the Følner functions of the (composed)
wreath products more general than (F• )?
75
Remarks. (a) The question of evaluating the Følner functions for wreath
products was raised by A.Vershik in [29]. C. Pittet and L. Saloff-Coste, found
(see [27]) a lower bound on the Følner function F◦ of the wreath products of
Abelian by finite groups (i.e. F inoAb ); their inequality, albeit asymptotically
non-sharp, provided the first examples of F◦ with super-exponential growth.
The Vershik question was answered in [9] with the inequality (F oF ), where
the iterated case followed by induction. (The constants
P ci and εi in ci Fi (εi ni )
in [9] are smaller than those imposed by the above i (εi + ci (1 − εi )) < 1 and
the inductive argument makes them dependent on the shape of bracketing with
the best result for (()...()) where the induction goes from i to 2i. We see the
relevance of the constants below.)
(b) The isoperimetric inequalities for iterated wreath products of finite groups
Γ and graphs (see [9]) and their modifications provide examples of graphs with
expander-type isoperimetry (see [19]).
(c) Wreath Products over Homogeneous Spaces. The above inequalities for
groups straightforwardly generalize (see [9]) to homogeneous spaces, e.g. to
2
X2 = Γ2 /Γ02 and the group Γ ⊃ ΓX
generated by the restricted Cartesian
1
X2
2
power group Γ1 , where Γ2 naturally acts on ΓX
1 :
2
the max-Følner function F◦ (n; Γ/Γ02 ) of the action of Γ on Γ/Γ02 = ΓoX
as
1
well as the max-Følner of the group Γ itself are bounded in terms of those for
Γ1 and the action of Γ2 on X by
F◦ (n; Γ) ≥ F◦ (n; Γ/Γ02 ) ≥ F2 · F1F2
for F1 (n) = 12 F ( n3 ; Γ1 ) and F2 (n) = 12 F ( n3 ; X2 ).
2
(d) The lower bound by (LoL
1 ) for E◦ -function of the wreath powers (of sets)
combined with 4.4 yields a proof of the following version of F oF for the Følner
functions associated to I◦incr , the maximal monotone increasing minorant of I◦ ,
F incr -Inequality.
incr
2
F◦incr (ΓoΓ
(Γ2 ,
1 , n) ≥ F◦
incr
n F◦ (Γ2 , n2 )
n incr
) F◦ (Γ1 , )
.
2
2
This inequality F incr is slightly sharper than F oF if Γi are (necessarily infinite) groups with monotone increasing profiles I◦ (r). But F oF , meaningfully
applies to finite groups where the inequality F incr is vacuous.
Isoperimetric Inequality for Groups with Distorted Wreath Product Subgroups..
Take an iterated wreath product Γ of Γi and let α be an endomorphism of Γ
induced by injective endomorpisms αi of Γi . Then Γi can be distorted in the
group Γα = Γ ×α Z.
For instance, if Γi are free Abelian and αi (γi ) = 2γi for all i and γi ∈ Γi ,
then Γi ⊂ Γα are exponentially distorted and, thus,
F◦ (n, Γα ) ∼ F◦ (exp(n), Γ).
9.3
Infinite o- and ↑-Products.
Probably, the Cartesian products and the wreath products are the only 2variable functorial operations, obtained by adding universal relations to free
products of pairs of groups, in the category of amenable groups, but there are
further ”products” in the category of ordered groups obtained as follows.
76
Denote by Γon = Γ(on the n-th wreath iteration .(.(.(... with all Γi = Γ1 , let I
be an arbitrary countable ordered set and assign to each n-element subset J ⊂ I
the group Γo1J = Γo1n . This assignment is obviously functorial for inclusions of
subsets and we can define the inductive limit for J exhausting I, denoted Γo1I .
If I is infinite then this group is infinitely generated; but if we take some
finitely generated group Γ2 of order preserving transformations transitively acting on I, then the group generated by Γo1I and Γ2 is finitely generated. If we
2
take an ordered group Γ2 for I, we denote the resulting group Γ by Γ↑Γ
(in
1
agreement with Knuth arrow notation) and regard it as the ”first order wreath
product” (we regard o as the zero order product) of the two groups. (See [9] for
the case I = Z).
(↑
We iterate this n times and, by passing from Γ↑1n = Γ1 n to the inductive
limit, arrive at next product Γ = Γ1↑↑Γ2 , where we need both groups to be
ordered. Similarly we define the ↑↑↑-product ,etc.
The above construction equally applies to the right bracketing .).).)... but
it does not (seem to) work as it stands for the ...(()())...-bracketing: we need a
finitely generated order preserving group Γ2 transitive on all n-tuples of points in
I. There is a finitely generated group with this property, the Richard Thompson
group, but this group is unknown to be amenable or non-amenable. (We are keen
on getting large amenable group by such procedures). On the other hand the
finitely generated group obtained in such a way is rather amusing; for instance,
it contains the finite iterated wreath products with all positions of the brackets
and the same applies to ↑↑ ...-products of any given order.
Questions. (a) Do the E◦ -functions (and/or the Følner functions) of the
partitions of the above Γ into the orbits of Γi equal (asymptotically equivalent
to) the corresponding ”arrowings” of the function n1 o n2 = n2 nn1 2 ?
Here is a (little) step in this direction.
Isoperimetric Inequality for Γ = Γ1 ↑ Γ2 . Let Γk ⊂ Γ be the subgroup
generated by the union of Gγ1i ⊂ Γ, i = 1, 2, ..., k, for some k elements in Γ2 of
the G2 -length ≤ l, where Gi are given generating subsets in Γi ⊂ Γ, i = 1, 2.
The max-Følner function of the group Γk , that isomorphic to the left-iterated
i.e. ...o(o(..., bracketed wreath product of Γ1 with itself, is bounded from below by
the k-iterated o-composition of Fk (n) = 1/2kF◦ (n/2k; Γ1 , G1 ), denoted ok Fk (n),
where the max-boundary refers to all k|G1 | generators in Γk ⊂ Γ.
Thus, the Følner function of (Γ, G1 ∪G2 ) is bounded from below by Fk (n/3l)
(compare with 6.1).
Take γi from Gl2 ⊂ Γ2 , such that l · |Gl2 | ∼ n/10 for the appropriate l =
l(n), and regard the cardinality |Gl2 | as the function ϕ(n). Form example, if the
growth function G◦ (l) = |Gl2 | of Γ2 equals lm then ϕ(n) ∼ n m+1 and if Γ2 has
exponential growth, then ϕ(n) ∼ n/log(n).
Then, by the argument in in 6.1
the Følner function F◦ (n; Γ) is bounded from below by oϕ(n) 2n that is equivalent to the ϕ(n)-iterated exponential function exp(exp(...(exp(2))...)) .
Questions. (a) Apparently, one has a poor understanding of the combinatorial structure of the orbit partitions of such groups as Γ1 ↑i Γ2 , where the
attractive feature of this structure is being universal, defined for arbitrary ordered sets, rather than groups, and where the necessity of the order is due to
the fact that the o-product, a two-variable functor on sets, is non-commutative.
If i = 2, there are two partitions P1 and P2 of Γ1 ↑ Γ2 , where the combina-
77
torics is faithfully encoded by the bipartite graph on the vertex set of the slices
of Pi with the edges corresponding to the pairs of intersecting slices.
What is the combinatorics of such graphs? For example, what are the spectra
(of the combinatorial Laplacians) of such graphs constructed with finite ordered
sets?
Does the graph theoretic picture reveal anything or it is better to think in
terms of partitions?
What does correspond to the concepts ”measure” and/or ”entropy” that
would lead to something like Shannon inequalities?
Do the groups like Γ1 ↑i Γ2 embed (preferably functorially) into finitely
presented ordered groups Γ (similar to Thomson group Γ ⊃ Z o Z), where one
also wants Γ to be amenable for amenable Γi ?
Are there other ”interesting ” milti-variable operations (functors?) in the
category of finitely generated groups that preserve the classes of amenable
and/or of orderable groups? What are the functors from the category of finite ordered sets to the category of such operations on groups?
An example of a one variable operation is Γ
Γ0 ⊃ Γ, where Γ0 consists
of the bijective maps Γ → Γ which are equal to left translations in Γ away
from a finite subset. (If G ⊂ Γ generates Γ then Γ0 is generated by G and the
permutations of G; probably, F◦ (Γ0 ) ∼ (F◦ (Γ))!.)
There are similar extensions of Γ with ”simple” bijective maps Γ → Γ, e.g.
the maps that are translations on a subgroup Γ0 ⊂ Γ and that fix γ ∈ Γ \ Γ0 ,
but making such an extension functorially is problematic.
(b) It seems impossible to continue the ↑ hierarchy in the category of ordered
groups. For example, it is unclear if all groups Z(↑k embed into an amenable
group on two generators. (In general, one does not seem to know which countable families of finitely generated amenable groups embed into a single finitely
generated amenable group, while the answer is an easy ”yes for all” in the class
of all groups by the small cancellation theory for relatively hyperbolic groups.)
Yet, one can go several steps up if the background group Γ2 admits an infinite
strictly increasing family of subgroups say Γ3 ⊂ Γ4 ⊂ ... ⊂ Γk ⊂ .... where,
similarly to the k-commutator relations from in 8.2, one adds the relations
defining the kth arrowing operation to each Γk . Then the resulting group Γ
(that can be arranged with three generators) contains all Z(↑k and, probably,
its Følner function grows (at least) as fast as the Ackermann function.
It is unclear up to which ordinals one can go in a similar manner keeping
groups finitely generated. (Every elementary amenable group Γ built by normal
extensions and inductive limits is characterized by the ordinal that expresses
the minimal ”number” of steps in such description of Γ; it seems unknown
if all countable ordinals correspond to finitely generated Γ). Apparently, one
needs a group theoretic realization of more powerful (implicit) recursion schemes
starting from groups with some ”self similarity”, e.g. finitely generated (even
better finitely presented ) amenable groups Γ containing (better, isomorphic to)
ΓoΓ ; but such groups are yet to be found (or shown not to exist).
10
Brunn-Minkowski inequalities.
Recall that the Euclidean Brunn-Minkowski inequality provides the following
lower bound on the product (called Minkowski sum and usually denoted ”+”
78
rather than ”·” in the Abelian case) of measurable subsets Y1 , ..., Ym in Rk ,
1
BM k
1
1
1
1
|Y1 · Y2 · ... · Ym | k ≥ |Y1 | k + |Y2 | k + ... + |Ym | k .
(See the survey article by Gardner [12].)
1
Remark on Nilpotent Groups. The classical proof of BM k extends to the
1
nilpotent case and yields BM k in the simply connected nilpotent (and, probably,
solvable) Lie groups Γ of dimension k.
10.1
Asymptotic Brunn-Minkowski inequalities for Discrete Groups.
1
The inequality BM k , unlike the isoperimetric inequality, does not directly pass
to co-compact discrete subgroups Γ as it is seen, for example, for subsets Y1 and
Y2 in Γ = Z2 that are both contained in a cyclic subgroup. To correct for this
one may stabilize the Y ’s by taking their powers
Y N = |Y · Y {z
· ... · Y} ⊂ Γ.
N
(Such stabilization in the context of polynomial rings was suggested to me by
Andrei Suslin.)
If Γ = Zk then the majority of points in the power set Y N of every finite
subset Y ⊂ Zk are obtained by intersecting the subgroup ΓY ⊂ Γ generated by
Y with a convex subset CN ⊂ Rk ⊃ Zk , where the subsets N1 CN converge in
Rk for N → ∞.
1
Then BM k in Rk implies the following asymptotic inequality in Zk ,
1
BMεk
1
1
1
1
(|Y1N · Y2N · ... · YmN |) k − (|Y1N | k + |Y2N | k + ... + |YmN | k ) ≥ −εN N k
where, εN → 0 for N → ∞.
Question. What happens for finite N ? Is there a meaningful bound on |εN |
independent of Y ’s ? Are there BM-type inequalities between ”monomials”
|Y1N1 · Y2N2 · ... · YmNm |?
Naively, one expects that BM-type lower bounds on |YiN |, say |Yi2 | ≥ 2k |Yi |,
may yield such a bound on |Y1 · Y2 · ... · Ym | but it seems rather subtle already
for Z.
1
Limit form of BMεk for N → ∞. Since Zk is commutative, so is the
Minkowski sum (written here as product) and Y1N · Y2N = (Y1 · Y2 )N . Us1
ing this and BMεk , one concludes that, for every finite Y ⊂ Zk there exists the
limit,
|Y |∞ =def limN →∞
1
|Y N |,
Nk
1
and the inequality BMεk takes a more compact form,
1
k
BM∞
1
1
1
1
k
k
k
k
|Y1 · Y2 · ... · Ym |∞
≥ |Y1 |∞
+ |Y2 |∞
+ ... + |Ym |∞
.
79
Remark. Notice that this inequality, unlike the Euclidean BM does not imply
the isoperimetric inequality for general subsets Y ⊂ Γ since the stabilization
renders subsets essentially convex.
Nilpotent groups. If Γ is a torsion free nilpotent group with the growth
function G◦ (n) ∼ nl , then the above argument yields,
1
BMεl
1
1
1
1
(|Y1N · Y2N · ... · YmN |) l − (|Y1N | l + |Y2N | l + ... + |YmN | l ) ≥ −εN N l ,
where εN → 0 for N → ∞.
1
k
Questions (a) Is there some version of BM∞
for nilpotent groups?
In general, let Γ be any countable group, Yi , i = 1, 2, ...m, finite subsets in
1
2
n
Γ and consider all ”non-commutative monomials” |YiN
· YiN
· ... · YiN
| for all
1
2
n
n = 1, 2, .... (This |... · ...| for fixed Yi , i = 1, ..., m, makes a submultiplicative
function on the free semigroup with m generators.)
(b) What are Γ-universal BM-type inequalities between these monomials?
(Here ”universality” allows dependence of these inequalities on Γ and Nj but not
on Yi except for some mild conditions on Yi such as being symmetric generating
subsets in Γ, where such an assumption simplifies the picture already in the
Abelian and nilpotent cases.)
Example. Let Γ be a finitely generated group of exponential growth and set
1
log|Y N |.
N
(c) When does the following asymptotic equality hold true?
|Y |log = limN →∞
(=)
limN →∞
1
log(|Yia1 1 N · ... · Yianm N |) = a1 |Yi1 |log + ... + am |Yin |log .
N
This is tautologically true if all Yi are equal and it is also obvious for pairs of
subsets Y1 and Y2 in the free group Fk , where Y equals the standard symmetric
generating set (with 2k + 1 elements) in Fk .
Also, (=) also holds true for symmetric generating sets Yi in hyperbolic
groups Γ. For example let m = 2, regard Y1N and Y2N as N -balls with respect to the word metrics associated with the generating sets Y1 and Y2 and
think of the product Y1N · Y2N as the N -neighborhood the first ball in the metric
defined by Y2N .
Both balls are quasiconvex and the rate of growth of the N -neighborhoods
of a quasiconvex set U ⊂ Γ for every word metric is proportional to the rate
of growth of the balls themselves, while the initial condition for the growth is
given by the cardinality of the boundary of U ,
|U · Y2N | ∼ |∂Y2 (U )| · |Y2N |.
This is seen with the finiteness (Markov) property of the boundary types of
boundaries of balls in hyperbolic groups.
In our case of U = Y1N the cardinality of the boundary of U is roughly the
same as the cardinality of U itself; this concludes the proof in the present special
case and the general case goes along the same lines.
Remark. There is a parallel discussion for probability measures on Γ instead
of of subsets, where the Minkowski products are replaced by convolutions of
80
measures and the cardinalities of subsets by the entropies of these measures,
but I have not worked out any example of a BM-inequality in this context.
10.2
Linearized BM-inequalities for Biorderable Groups.
Given an algebra A over some field F and linear subspaces D1 , D2 ⊂ A we denote
by D1 ∗ D2 ⊂ A the F-linear span of the products d1 d2 for all d1 ∈ D1 and d2 ∈
D2 , and we reiterate the above discussion with the D-monomials
1
2
n
|DiN1 1 ∗ DiN2 2 ∗ ... ∗ DiNnn | instead of |YiN
· YiN
· ... · YiN
|,
1
2
n
where |...| in the D-context stands for the F-rank of a linear space rather than
cardinality as in the Y -situation.
We are most interested in the case of A being the group algebra of some Γ
where we expect the D-monomials satisfy BM-type inequalities similar to those
for Y -monomials and we shall prove this below for biorderable groups:
a group Γ is called biorderable if it admits an order invariant under left and
right translations.
Examples. (a) Abelian groups without torsion are, obviously biorderable.
(b) Central extension of biorderable groups by biorderable are, clearly, biorderable.
(c) Inductive and projective limits of biorderable groups are,obviously, biorderable.
It follows that the residually torsionless nilpotent groups are biorderable. In
particular free groups are biorderable. Also the fundamental groups of surfaces,
except for the projective plane and the Klein bottle, are bi-orderable. (The
fundamental group of the Klein bottle is, clearly, left orderable. See [25] for
more geometric examples of (bi)-orderable groups).
for functions f on ordered
Recall (see 3.1) the correspondence f 7→ xmin
f
sets X and observe that
xf ∗g = xf · xg
for X being a biordered group Γ and ∗ denoting the (convolution) product in
the group algebra A of F-valued functions with finite support on Γ.
It follows that the assignment
A ⊃ D 7→ YDmin ⊂ Γ
(of the sets of non-zero ”diagonal entries” of triangular bases to finite dimensional subspaces D ⊂ A) satisfies
YDmin
⊃ YDmin
· YDmin
.
1 ∗D2
1
2
(This was pointed out to me by Dima Grigoriev for polynomial rings with
the lexicographic order on monomials.)
Therefore,
If Γ is a biorderable group (or semigroup) then its group algebra A over
any field has the same BM-profile as Γ: every BM-type inequality between Y monomials passes to the corresponding inequality for D-monomials.
Then, by combining this with the asymptotic Brunn-Minkowski inequality
1
k
BM∞
for a free Abelian (or nilpotent) group Γ, we arrive at
Linearized BM-Inequality for the Group Algebra A of Γ.
81
1
1
1
1
1
k
k
k
k
|D1 ∗ D2 ∗ ... ∗ Dm |∞
≥ |D1 |∞
+ |D2 |∞
+ ... + |Dm |∞
.
k
BMlin
Remark. The Euclidean BM-inequality, as well as a more powerful AlexandrovFenchel inequality for mixed volumes for convex sets, can be proved by an
algebra-geometric (or by an essentially equivalent complex analytic) argument
due to Hovanski and Teissier (see [13] and references therein), where BM is reduced to a similar inequality between the ranks of the spaces H 0 (Li ) of regular
sections of positive line bundles Li , i = 1, 2, ..., m, (over toric varieties associated
to the sets of the integer points of convex polyhedra) and the tensor products
1
k
(and also Alexandrov-Fenchel) remains valid, under
of Li . Probably, BMlin
suitable assumptions, for linear subspaces Di ⊂ H 0 (Li ) and it would be also
interesting to find something similar for non-commutative nilpotent groups Γ.
10.3
BM-Inequality for Fast Decaying Functions on Γ.
Since the construction of the triangular bases needs only the one-sided bound
on the support of functions on Γ = Zk , the above argument shows that
1
k
the inequality BMlin
holds in the (commutative) algebra A∞ of formal power
series in k variables over any field.
It follows that
1
k
(1alg ) The inequality BMlin
holds true in the algebra of regular functions on
an arbitrary irreducible k-dimensional algebraic variety,
1
k
(2C ) BMlin
holds for finite dimensional linear spaces Di in the algebra A of
C-valued function on Zk with exponential decay at infinity (relative to the word
length in Zk ), where the product in A is given by the convolution.
In fact, such functions are Fourier transforms of real analytic functions f on
the torus Tk the above applies to the Taylor expansions τt (f ) of f at some point
t ∈ T.
1
k
Remarks. (a) The super-polynomial rate of decay is not sufficient for BMlin
in (2C ), since the ring of C ∞ -function on Tk has zero divisors.
(b) One needs significantly less from an algebra A of smooth functions than
1
k
analyticity in order to have BMlin
. For example, if the Taylor map τt : A → A∞
for f 7→ τt (f ) is injective, i.e. if τ (f ) 6= 0 unless f = 0 (at least in a neighborhood
1
k
holds in A.
of t ∈ T ), then BMlin
1
k
Moreover, BMlin
holds in algebras A of quasi-analytic functions, where the
germs of function at a point t can be graded by ordinal numbers but I do
not know the most general condition on the (relative or absolute) decay rate
that would ensure the injectivity of τ (extended to infinite ordinals) and/or the
1
k
inequality BMlin
.
Remark. Linear spaces D of C-valued functions on a general group Γ with
sufficiently fast (superexponential ?) decay are likely to satisfy the same type
of BM-inequalities as spaces of functions with finite supports, but the approximation argument from 1.7 does not seem to apply in the present case.
1
k
3Qp The inequality BMlin
holds true for the decaying F-valued function f :
Γ → F for an arbitrary discrete valuation field F , e.g. for F = Qp .
Proof. Let f (x) be a non-zero analytic function that is a power series,
82
f (x) =
X
ai xi for ai ∈ F
−∞<i<+∞
with decaying coefficients, ||ai || → 0 for i → ±∞, and let m(f ) denotes the
number of the coefficients ai with the maximal norm, i.e. with ||ai || = ||f || =def
maxi ||ai ||.
If f = (1 − x)d g(x) for an analytic function g, then ||g(x)|| ≤ ||f (x)|| and
d ≤ m(f ) − 1.
P
In fact, the coefficients bi of ( i ai xi )/(1 − x) are the partial sums,
X
bi =
aj
j≤i
and if bj → 0 for j → +∞, then m(f (x)/(1 − x)) ≤ m(f ) − 1; thus, at most
m(f ) − 1 divisions are possible in the rings of analytic functions, since ||(1 −
x)d || ≤ ||(1 − x)||d = 1 and ||(1 − x)d g(x)|| ≤ ||g(x)||.
This implies the (trivial special case of)
Division Theorem. Each non-zero analytic function f admits a unique
product decomposition
f (x) = (1 − x)d g(x), where g(x) is an analytic function
P
with g(1) = i bi 6= 0.
Next, consider a central extension
1 → Z → Γ → Γ1 → 1
and observe (this was explained to me by Ofer Gabor) that this division
P automatically extends to the algebra A(Γ) of ”F-analytic functions” f = γ∈Γ a(γ)
with decaying coefficients a(γ), where the role of x ∈ A is played by one of the
two generators +1 or −1 in Z ⊂ Γ ⊂ A, say by +1:
every non-zero f ∈ A(Γ) is uniquely decomposable as f = (1 − x)d g, where
g ∈ A(Γ) is such that its image in A(Γ1 ) under the push forward homomorphism
does not vanish.
This implies, in particular, by induction on k that the algebra A(Zk ) embeds
(by the counterpart of the above τ -homomorphism) into the algebra of formal
power series and the proof of 3Qp follows.
Remark on non-Commutative Groups Γ. Let I ⊂ A denote the augmentation
ideal (generated by 1 − gj for some generators gj ∈ Γ) of the group algebra A
of Γ and let Ai denote the quotients A/I i for the powers I i of I.
These → Ai → Ai−1 → ... → A1 = F make a projective system of nilpotent
algebras that are finite dimensional for finitely generated Γ, where the projective
limit of this system is denoted by A∞ and is equipped with the usual topology.
If Γ = Zk , then A∞ equals the space of formal power series (in k commuting
variables) at the point t = 1 = (1, 1, ..., 1) in the torus Tk .
More generally, if the intersection of I i equals zero (e.g. if Γ is a torsion free
nilpotent or a free group), then A∞ is regarded as the formal completion of A.
The above division argument shows that the algebra A(Γ) of F-analytic functions embeds to A∞ and a similar conclusion holds for Archimedean fields (e.g.
R and C) for ”analytic” functions with a suitably defined radius of convergence
> 1. Thus the isoperimetric problems in A(Γ) reduce to these in A∞ .
Question. When does the algebra A∞ satisfy the same isoperimetric and
BM inequalities as A itself? Is it true in the case where the intersection of
83
the powers of the ideal I equal zero? (This ”nil-intersection” condition is quite
strong: if the underlying field F has characteristic zero it is close to the group
being residually nilpotent and for a finite field it looks even more restrictive.)
11
Bibliography.
References
[1] Almgren, Jr. F.J., Optimal isoperimetric inequalities, Indiana Univ. Math.
J., 35 (1986), 451-547.
[2] Bartholdi,
L.
On
Amenability
of
Group
arXiv:math/0608302v2, to appear in Isr. J. Math.
Algebras,
I,
[3] Belov, A., Mikhailov. R. Free Algebras of Lie Algebras Close to Nilpotent,
Preprint, 2008.
[4] Coulhon, T., Saloff-Coste, L., Isoprimtrie pour les groupes et les varits.
(French) [Isoperimetry for groups and manifolds] Rev. Mat. Iberoamericana
9 (1993), no. 2, 293–314.
[5] Effros, E.G., New Perspectives and some Celebrated Quantum Inequalities.
(2008) arXiv:0802.0006v2[math-ph].
[6] Elek, G. The amenability and non-amenability of skew fields. Proc. Amer.
Math. Soc. 134 (2006), no. 3, 637–644.
[7] Elek, G. The amenability of affine algebras. J. Algebra 264 (2003), no. 2,
469–478.
[8] Elek, G. On the Analytic Zero Divisor Conjecture of Linnell Bulletin of the
London Mathematical Society 2003 35(2), 236-238.
[9] Erschler, A. On isoperimetric profiles of finitely generated groups. Geom.
Dedicata 100 (2003), 157–171.
[10] Erschler, A. Piecewise automatic groups. Duke Math. J. 134 (2006), no. 3,
591–613. 20F65
[11] Friedgut, E. Hypergraphs, Entropy and Inequalities. The American Mathematical Monthly, 111 (2004), no. 9, 749–760.
[12] Gardner, R. J. The Brunn-Minkowski inequality. Bull. Amer. Math. Soc.
(N.S.) 39 (2002), no. 3, 355–405.
[13] Gromov, M. Convex sets and Kahler manifolds, in Advances in J. Differential Geom., F. Tricerri ed., World Sci., Singapore. (1990), 1-38.
[14] Gromov, M. Topological Invariants of Dynamical Systems and Spaces of
Holomorphic Maps: I - Mathematical Physics, Analysis and Geometry,
vol. 2, n. 4(1999), 323-415.
[15] Gromov, M. Random walk in random groups, GAFA, Geom. Funct. Anal.
13 (2003), no. 1, 147-177.
84
[16] Gromov, M. Singularities, Expanders, Hyperbolic Geometry and Fiberwise
Homology of Maps, In preparation.
[17] Gromov, M. Mendelian Dynamics and Sturtevants Paradigm, to appear
in Contemporary Mathematics Series, Geometric and probabilistic Structures in Dynamics, Keith Burns, Dmitry Dolgopyat, Yakov Pesin Editors.
American Mathematical Society, Providence RI1.
[18] Gromov, M., Milman, V. The spherical isoperimetric inequality in Banach
spaces, Compositio Math. 62 (1987), 263-282.
[19] Hoory, S.,Linial, N. and Wigderson, A. Expander graphs and their applications. Bulletin of the AMS, 43(4):439-561, October 2006.
[20] Lanford, O.E. Entropy and equilibrium states in classical statistical mechanics, in Statistical Mechanics and Mathematical Problems, Vol. 20 of
Lecture Notes in Physics, edited by A. Lenard (Springer-Verlag, Berlin,
1973).
[21] Mishor, P.W., Petz, D., Anda, A., Infinite Dimensional Analysis and Related Topics, Vol. 3, No. 2 (2000) 199-212.
[22] Morris, D.W. Left-Orderable Amenable Groups are Locally Indicable, Algebraic and Geometric Topology, v,6(2006) 2509-2518.
[23] Ohya, M., Petz, D. Quantum Entropy and Its Use, Series Theoretical and
Mathematical Physics. Springer, 2004.
[24] Petz, D., Sudar, Cs., Extending the Fisher Metric to Density Matrices, in
Geometry of Present Days Science, es Barndoff-Nielsen and Jensen, 21-34,
World scientific, 1999.
[25] Rolfsen, D. Ordered Groups and Topology. Lecture Notes, Luminy, June
2001.
[26] Ruskai, M.B. Another Short and Elementary Proof of Strong Subadditivity
of Quantum Entropy. Rep. Math. Physics 60 (2007), no. 1, 1-12.
[27] Pittet, C. and Saloff-Coste, L. Amenable groups, isoperimetric profiles and
random walks. in Geometric Group Theory Down Under (Canberra, 1996),
J. Cossey, C. F. Miller III, W. D. Neumann, and M. Shapiro, Eds., pp.
293316, de Gruyter, Berlin, 1999.
[28] Varopoulos, N. Th., Saloff-Coste, L., Coulhon, T. Analysis and geometry
on groups. Cambridge Tracts in Mathematics, 100. Cambridge University
Press, Cambridge, 1992.
[29] Vershik, A. Amenability and approximation of infinite groups. Sel. Math.
Sov. 2, 311-330 (1982).
85